Universal transport coefficient behaviour in ionic melts

Article abstract of DOI:10.1088/0953-8984/11/45/302

Precise resistivity and viscosity measurements for ionic mixed melts, regular and glass-forming, have been performed on the liquid state over a wide range of viscosity variation. Their universal behaviour in the high-temperature region allows us to scale them relative to their activation energy. The resistivity-viscosity relation reveals a fractional-power character over the whole range of parameters investigated. This interdependence suggests a fractional character of the Stokes-Einstein law of the viscosity-diffusion relation for ions in melts.

Full text of DOI:10.1088/0953-8984/11/45/302

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Universal transport coefficient behaviour in ionic melts
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J. Phys.: Condens. Matter 11 (1999) 8773–8784. Printed in the UK
PII: S0953-8984(99)05577-0
Universal transport coefficient behaviour in ionic melts
E Veliyulin†, E Shasha†, A Voronel†, V Sh Machavariani†, Sh Seifer†,
Yu Rosenberg‡ and M G Shumsky§
† School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences,
Tel-Aviv University, 69978, Tel-Aviv, Israel
‡ Wolfson Centre for Materials Research, Tel-Aviv University, Ramat Aviv 69978, Israel
§ Graduate Center for Materials Research, University of Missouri–Rolla, Rolla, MO 65401, USA
Received 28 June 1999
Abstract. Precise resistivity and viscosity measurements for ionic mixed melts, regular and glass-
forming, have been performed on the liquid state over a wide range of viscosity variation. Their
universal behaviour in the high-temperature region allows us to scale them relative to their activation
energy. The resistivity–viscosity relation reveals a fractional-power character over the whole range
of parameters investigated. This interdependence suggests a fractional character of the Stokes–
Einstein law of the viscosity–diffusion relation for ions in melts.
1. Introduction
In our previous papers [1, 2] a universal behaviour of the viscosity for a variety of ionic
melts was demonstrated over a wide range of parameters. Although the conductivity has been
extensively measured before [3], we still lack systematic data for a wide range of temperature to
compare with the viscosity. While for the usual simple melts the resistivity seems to be roughly
proportional to the viscosity, this ceases to be the case over the wider range of parameters which
is achievable for mixed melts. We have performed a series of precise resistivity measurements
for ionic mixed melts with low melting points to allow comparison with the corresponding
viscosity data. In particular, mixtures exhibiting a trend to glassification have been chosen, to
considerably stretch the range of the parameters achievable within the liquid state. The nitrate
melts NaNO3, RbNO3, KNO3, Ca(NO3)2 and their mixtures exhibit viscosities amenable
to precise measurement over a wide but convenient range of temperature. Mixing these
salts enables one to prepare samples of different glass-forming abilities depending on the
composition. This also provides the opportunity to compare the results with our previous
viscosity and optical measurement data [2, 4]. For comparison, we have also measured the
resistivityofa(KCl)0.33(AlBr3)0.67 mixtureandaglass-formingLiCl–watersolution. Avariety
of data from the literature [5–7] have been considered, to test the generality of our approach.
2. Experimental procedure
Ultrapure KNO3 and Ca(NO3)2·4H2O (Alpha Corporation), NaNO3 (99.995%), RbNO3
(99.7%) and AlBr3 (99.99%) (Aldrich Corporation), LiCl (99+%) (Merck) and KCl (99.9%)
(Johnson–Matthey) components were used for sample preparation. In order to obtain pure
Ca(NO3)2 salt, crystallohydrate crystals of Ca(NO3)2·4H2O were slowly heated under vacuum
0953-8984/99/458773+12$30.00 © 1999 IOP Publishing Ltd
8773

8774
E Veliyulin et al
up to 150 ◦C and kept at this temperature for more than five hours. The dehydration procedure
was additionally controlled by sample weight registration. The alkali nitrate crystals were
dried under vacuum at 150 ◦C for several hours. The dry components NaNO3, KNO3, RbNO3
and Ca(NO3)2 were treated in a glove box under a controlled nitrogen atmosphere, weighed
out and mixed. The ready-mixed sample was put into a hermetically closed measuring cell.
Our resistivity measurements were performed by a two-terminal AC technique with a
Hewlett-Packard 4284A precision LCR meter (20 Hz–1 MHz). The lower part of the meas-
uring cell consists of a calibrated glass tube (with an internal diameter 0.339±0.001 cm) with
two welded electrodes (0.3 mm diameter Pt wire) at a distance 8.00±0.05 cm from each other.
After melting it within the measuring cell, the sample was kept at the highest-temperature
point in a liquid state for about ten hours to enable complete mixing of the components
and, thus, to ensure the stability of the measured impedance. Then it was cooled down at a
constant rate (50 K h−1) and the impedance was measured as a function of temperature at seven
different frequencies (1, 5, 10, 50, 100, 500 and 1000 kHz). The system was fully monitored
by a personal computer through the GP-IB bus and RS-232 serial port. The temperature was
measured by a Pt versus Pt + 10%Rh thermocouple with a reference junction and controlled
by a Eurotherm programmable controller.
An equivalent circuit consisting of four elements was constructed (figure 1) to simulate
the measured impedance. These four elements are: a resistor Rs simulating the DC bulk
resistance of a sample; a contact capacitance Ccont simulating the AC electrode polarization
effect; and two parasitic elements—a coil Lp and capacitance Cp—representing the inductance
and capacity of the measuring cell itself. The bulk resistance of the sample Rs was extracted
from the measured impedance by solving numerically a set of equations derived from the
proposed circuit of figure 1 for (at least) five different frequencies. The resistivity was deduced
afterwards from the resistance data and the geometrical parameters of the cell.
RS
Ccon
Lp
Cp
Figure 1. The equivalent circuit of the resistivity measurement scheme: Rs is a resistor simulating
the DC bulk resistance of the sample; Ccont is a contact capacitance simulating the AC electrode
polarization effect; Lp and Cp are the inductance and capacitance of the measuring cell itself.
3. Phase diagrams
Knowledge of a mixture’s phase diagram is important because it makes it possible to determine
the positions of the melting points, and therefore also the crystallization trend of the sample,
in a preliminary fashion. As a rule, the diagrams for the states reported by reference [3] are
reliable. However, the RbNO3–NaNO3 diagram reported in [3] to exhibit unlimited solubility
in the solid state looks doubtful.
Indeed, our thermophysical and our x-ray diffraction studies gave evidence of a different
kind of phase diagram for RbNO3–NaNO3, which is presented in figure 2. The liquidus line
temperatures (and in fact those of the solidus line too, but with less certainty) were detected
from abrupt changes of density [1] and conductance for different compositions of this system.

Universal transport coefficient behaviour in ionic melts
8775
600
550
T , [K]
500
data from [3]
data from resistance (liquius line)
data from resistance (solidus line)
data from density change detection
450
0.0
RbNO3
0.2
0.4
0.6
0.8
1.0
Mole % of NaNO
NaNO
3
3
Figure 2. The phase diagram of RbNO3–NaNO3. The inset shows schematically the form of the
diagram in the region of concentrations from 0.28 to 0.43 for NaNO3.
The x-ray diffraction study of the solid–liquid phase transition for the Rb0.6Na0.4NO3
composition was performed with Cu Kα radiation on a 2:2 ‘Scintag’ powder diffractometer
equipped with a liquid-nitrogen-cooled intrinsic-Ge solid-state detector. In figure 3, one can
3000
2000
1000
0
4000
3000
2000
1000
0
T=297 K (right scale)
T=418 K (left scale)
Figure 3. X-ray diffraction patterns for
the Rb0.6Na0.4NO3 composition at different
temperatures: the upper part which cor-
responds to T = 297 K coincides with the
pattern of the stoichiometric crystalline phase
Rb2Na(NO3)3 (see reference [8]); the lower
15 20 25 30 35 40 45 50 55 60
part corresponds to the liquid state at T
418 K.
=
Diffraction angle 2Θ, degrees

8776
E Veliyulin et al
see the two diffraction patterns obtained from Rb0.6Na0.4NO3 at about its melting (supercooled)
temperature and at room temperature. While the former indicates the undercooled liquid state,
the latter pattern clearly indicates the existence of a new stoichiometric phase, Rb2Na(NO3)3,
which has been reported once before as a result of a hydride solution investigation in [8]. The
presence of this stoichiometric compound confirms the existence of a maximum between the
two minima at the bottom of the diagram in figure 2. Thus, in spite of the fact that RbNO3 and
NaNO3 are no different as regards their cationic charge, their mixture probably contains some
complex ions.
4. Resistivity data
Thepreciseresistivitymeasurementswereperformedfornitratemeltsintherangefromroughly
740 K down to the crystallization points (about 400–500 K). In the cases of glass-formers,
we were able to extend our to measurements well below this. For LiCl–water solutions, a
few concentrations were investigated in the range from room temperature down to roughly
180 K.
To analyse our resistivity data, one can use an exponential formula (the so-called Arrhenius
formula), as was done before for our viscosity data [1,2]:
ρ/T = Bρ exp(Eρ/T ).
(1)
Table 1 contains the parameters of equation (1) together with the values derived in an
analogous way from the viscosity data [1, 2]. Although in a low-temperature region the
data deviate considerably from the Arrhenius form, it still makes sense to use this form,
since the activation energy Eρ (in kelvins) provides a reasonable temperature scale for the
melts investigated. Figure 4 illustrates this for many melts (including ZBLAN20: 0.53ZrF4–
0.20BaF2–0.04LaF3–0.03AlF3–0.20NaF; see reference [7]) in an Arrhenius plot (lg(ρ/BρT )
versus Eρ/T ). Unfortunately, for both (KCl)0.33(AlBr3)0.67 and the LiCl–water solutions there
is no high-temperature region, but the ‘corresponding-states law’ based on scaling with Eρ is
still valid.
Table 1. Parameters of the Arrhenius formula for the resistivity (Bρ and Eρ) and the viscosity
[1, 2] (Bη and Eη) high-temperature data. The ratio of activation energies Eρ/Eη is also presented.
The missing data for some of the melts have been taken from references [3, 5, 6, 7].
Salt
Bρ ( cm K−1
)
Eρ (K) Bη (cP K−1
)
Eη (K) Eρ/Eη
LiNO3
3.55 × 10−5
5.11 × 10−5
5.18 × 10−5
5.21 × 10−5
6.27 × 10−5
8.71 × 10−5
6.81 × 10−5
5.79 × 10−5
6.66 × 10−5
7.23 × 10−5
8.58 × 10−5
3.55 × 10−5
0.96 × 10−5
2200
2050
2230
2390
2470
2370
1830
2400
2720
2850
2580
1890
5130
4.59 × 10−5
2920
2570
2660
2930
2790
3360
2310
2710
(3130)
3400
0.75
0.80
0.84
0.81
0.89
0.71
0.80
0.89
(0.87)
0.84
NaNO3
6.14 × 10−5
5.21 × 10−5
3.95 × 10−5
5.59 × 10−5
2.73 × 10−5
8.86 × 10−5
5.48 × 10−5
(6.00 × 10−5
4.35 × 10−5
KNa(NO3)2
KNO3
RbNO3
CsNO3
AgNO3
Rb3Na2(NO3)5
Ca2K3(NO3)7
Ca2Rb3(NO3)7
Ca3Na1Rb6(NO3)13
(LiCl)0.16(H2O)0.84
ZBLAN20
)
(0.29 × 10−5
2.85 × 10−5
)
(2460)
5910
(0.77)
0.87

Universal transport coefficient behaviour in ionic melts
8777
In accordance with equation (1), the scaled conductivity (BρEρ/ρ) can be presented as a
universal function of a single variable T/Eρ only:
ꢀ
ꢁ
BρEρ
ρ
Eρ
T
Eρ
T
=
exp −
.
(2)
Figure 5, which presents the data as a function of the scaled temperature T /Eρ, illustrates
the universality of equation (2) rather convincingly. However, in the low-temperature region
(where, roughly, T/Eρ < 0.2) the glass-formers deviate from the universal graph. One can
accept this as a definition of ‘high’- and ‘low’-temperature regions.
0.16
Ca2Rb3(NO3)7
108
Ca3Na1Rb6(NO3)13
Ca2Rb3(NO3)7
Ca3Na1Rb6(NO3)13
0.14
Ca2K3(NO3)7
LiNO3
Ca2K3(NO3)7
LiNO3
107
106
105
104
103
102
101
CsNO3
0.12
0.10
0.08
0.06
0.04
0.02
0.00
KNO3
KNa(NO3)2
NaNO3
CsNO3
KNO3
Rb3Na2(NO3)5
RbNO3
KNa(NO3)2
NaNO3
(LiCl)0.16(H20)0.84
(KCl)0.33(AlBr3)0.67
ZBLAN20
Rb3Na2(NO3)5
RbNO3
3
4
5
6
7
8
9
10
11
0.12
0.16
0.20
0.24
0.28
0.32
E /T
T/E
ρ
ρ
Figure 4. The scaled resistivities of ionic melts as Figure 5. The scaled conductivities of ionic melts as
functions of the inverse scaled temperature in a semi-
logarithmic plot.
functions of the scaled temperature.
The experimental conductivity data above T /Eρ > 0.2 look like a linear function of
temperature. This is not in contradiction with equation (1). Indeed, the function on the right-
hand side of equation (2) has an inflection point at
√
T/Eρ = (2 − 2)/2 ≈ 0.293.
Inthe vicinityofthispoint, the functionactuallybehavesalmost linearly. Thisleadsempirically
to the linear dependence of the conductivity on the temperature for the majority of melts (this
makes the experimental data compatible with ‘free-volume’ concepts [9]). It is particularly
the case for the nitrates, since the point T/Eρ = 0.293 corresponds for them to the middle of
the temperature range available for the measurements [10].
As one can see from figure 4, the data in the low-temperature region, halfway to
glassification, are not properly scaled by an Arrhenius formula with two fitting parameters.

8778
E Veliyulin et al
This is why many authors use a Vogel–Fulcher empirical formula with three par-
ameters [11] to present their data on glass-forming melts. The Vogel–Fulcher equation was
presented by Angell in the following form [12]:
ꢀ
ꢁ
DρT0ρ
ρ = Cρ exp
(3)
T − T0ρ
where Dρ is defined as a fragility coefficient and T0 is a characteristic constant identified by
some authors with the Kauzmann point [13]. The coefficient Dρ is greater for the stronger
glasses.
The parameters (Cρ, Dρ and T0ρ) of the Vogel–Fulcher equation for our resistivity data
over the full range of our measurements are presented in table 2.
Table 2. The parameters (Cρ, Dρ and T0ρ) of the Vogel–Fulcher equation for our resistivity data
(see equation (4)) together with the parameters (Cη, Dη and T0η) of the Vogel–Fulcher equation for
viscosity (see references [1, 2]). The parameters Dρ, Cρ and T0ρ for ZBLAN20 were obtained by
fitting over the high-temperature region (T > 500 K). The ratio of the fragility coefficients Dρ/Dη
is also presented. The missing data for some of the melts have been taken from references [3, 5, 6, 7].
Salt
Cρ ( cm)
Dρ
T0ρ (K)
Cη (cP)
Dη
T0η (K)
Dρ/Dη
Rb3Na2(NO3)5
Ca2K3(NO3)7
Ca2Rb3(NO3)7
Ca3Na1Rb6(NO3)13
Ca2Na2Rb6(NO3)12
(KCl)0.33(AlBr3)0.67
(LiCl)0.16(H2O)0.84
ZBLAN20
0.277
0.344
0.471
0.441
0.472
0.807
0.198
0.036
2.90
2.15
2.28
2.27
1.94
3.05
4.16
5.86
240
317
307
295
291
200
135
376
0.292
0.220
0.411
3.69
2.72
3.09
232
320
294
0.79
0.79
0.74
0.388
0.034
0.041
2.64
6.34
5.38
226
123
458
0.87
0.66
1.09
5. Viscosity data
The viscosity data for some of our melts were measured in earlier work [1, 2]. But CRN
(Ca2Rb3(NO3)7) glass-former viscosity measurements have been performed recently, for
comparison with our resistivity data. Figure 6 presents our viscosity data together with data
from the literature in an Arrhenius plot analogous to that of figure 4 for the resistivity. One can
see that ‘the corresponding-states law’ is valid also for viscosity if the temperature is similarly
scaled relative to the corresponding Eη. At low temperature the viscosity behaviour of the
melts is becoming material dependent, as also happens for the resistivity. Both the Arrhenius
and the Vogel–Fulcher equation parameters have been tabulated in tables 1 and 2. In the last
column of each table the ratios of the activation energies, Eρ/Eη, or fragility coefficients,
Dρ/Dη, are presented (Eρ and Dρ are derived from the resistivity data; Eη and Dη are derived
from the viscosity data). They appear to be in unexpectedly good agreement. However, a
comparison of figures 4 and 6 shows that one additional parameter Dη (‘fragility’) is not
enough to characterize the difference in glass-forming behaviour.
6. The resistivity–viscosity relation
The universal high-temperature behaviour of the resistivity (figure 4) and viscosity (figure 6),
on one hand, and the close ratios of the activation energies Eρ/Eη and fragility coefficients

Universal transport coefficient behaviour in ionic melts
8779
Dρ/Dη (table 2), on the other hand, allow us to check the resistivity–viscosity interdependence
straightforwardly.
As a result of our extended study of both electrical conductivity and viscosity, we are able
to consider the functional interdependence of these two quantities directly, without any use of
approximating formulae (Arrhenius or Vogel–Fulcher).
Figure 7 presents our experimental data for ionic melts in a log–log plot. The frequently
used combination of the Stokes–Einstein (SE) law [14, 15] for Brownian motion and the
Nernst–Einstein (NE) law [14,15] for ionic conductivity requires a linear resistivity–viscosity
dependence. Thus the slope of the resistivity–viscosity curve in a log–log plot is expected to
be equal to 1. However, the slope in figure 7 is in fact obviously less than 1 and roughly the
same (m ≈ 0.8 ± 0.1) for all of the melts investigated over the wide viscosity range starting
from viscosity 10−3 cP. Therefore the fractional conductivity–viscosity relation can be written
for a rather wide range of viscosity variation as follows:
ꢀ
ꢁ
m
ρ
T
η
T
∝
.
(4)
1011
1010
109
108
107
106
105
104
103
102
101
Ca2Rb3(NO3)7
Ca2K3(NO3)7
KNO3
Ca2Rb3(NO3)7
Ca2K3(NO3)7
NaNO3
101
KNa(NO3)2
NaNO3
Rb3Na2(NO3)5
(KCl)0.33(AlBr3)0.67
(LiCl)0.16(H2O)0.84
(LiCl)0.18(H2O)0.82
ZBLAN20
Rb3Na2(NO3)5
RbNO3
100
(LiCl)0.16(H20)0.84
ZBLAN20
RbI
RbCl
10-1
10-2
10-3
3
4
5
6
7
8
9
10 11 12 13
10-3
10-2
10-1
100
101
E /T
η/T [cP/K]
η
Figure 6. The scaled viscosities of the ionic melts as
functions of the inverse scaled temperature in a semi-
logarithmic plot.
Figure 7. ρ/T as a function of η/T in a log–log plot
for different ionic melts and solutions. The solid line
corresponds to the slope m equal to 1.
We note that the temperature factor T , which is insignificant for the glassification region, is
rather important for the high-temperature part of the dependence. The numerical value m from
equation (4) is in a good agreement with the values of the ratio Eρ/Eη derived for the high-
temperature region (see table 1). This confirms the intrinsic character of the high-temperature
parameters Eρ and Eη and equation (4).

8780
E Veliyulin et al
The phenomenological fractional relation (4) may be interpreted as a manifestation of a
fractional relation between the individual ion’s mobility and the viscosity of the medium as a
whole. This means assuming the fractional Stokes–Einstein (FSE) law to apply to microscopic
particles: D ∝ (η/T )m [16]. It agrees well with the straightforward measurements of the
translational diffusion coefficients of tracer molecules within viscous media [17, 18]. The
experimental data of [18] cover, in fact, 14 orders of viscosity variation but, unfortunately,
the authors used η as a variable rather than η/T , and thus received the impression that the
fractional law was valid only asymptotically close to the glassification limit.
Since for the index determination in a log–log plot the range of magnitude is particularly
important, the cases of the glass-formers Ca2K3(NO3)7 and the LiCl–water solution are
presented separately in figure 8, together with the experimental data on ZBLAN20 [7] melt.
This figure explicitly shows the non-linear character of the conductivity–viscosity relation
over fourteen orders of magnitude of viscosity variation. The exponent m varies in this range
between 0.9 and 0.5, decreasing in the glassification region.
108
107
106
105
104
103
102
101
Ca2Rb3(NO3)7
100
Ca2K3(NO3)7
Rb3Na2(NO3)5
10-1
(KCl)0.33(AlBr3)0.67
(LiCl)0.16(H2O)0.84
10-2
10-3
(LiCl)0.18(H2O)0.82
ZBLAN20
Figure 8. ρ/T as a function of η/T in a log–log plot
for different ionic melts and solutions for 14 orders of
magnitude of viscosity variation. The solid line cor-
responds to the slope m equal to 1.
10-2 100 102 104 106 108 1010 1012
η/T [cP/K]
Thus the log–log viscosity–conductivity plotting represents a both non-trivial and model-
independent way to characterize the transport properties of melts. While the exponent m in
equation (4) is not actually universal, its deviation from universality is material dependent.
One can expect new physical information to be derived from this dependence.
7. Discussion
The ‘corresponding-states law’ established in this work (see figures 4, 5 and equation (2))
allows us to properly define the ‘high’- and ‘low’-temperature regions. While the high-
temperature behaviour of melts can be scaled with energetic characteristics Eρ, the low-
temperature behaviour (T/Eρ 6 0.2) requires additional parameters.

Universal transport coefficient behaviour in ionic melts
8781
If one assumes the translational diffusion to be the main mechanism of charge transport in
the high-temperature region of a simple melt, the conductivity activation energy Eρ roughly
characterizes the mobility of the ions. However, an elementary viscous-flow process should
include more than an individual ion’s motion. Therefore, its activation energy Eη even at
high temperature cannot be the same as that for mass (or charge) transport. This leads to the
deviation of the exponent m from 1 in equation (4).
In spite of the fact that empirical data supporting equation (4) appeared as early as
1908 [19], the quasihydrodynamic approximation (m = 1) to molecular motion based on
the Stokes–Einstein formula for Brownian particles remained widely used [14]. Indeed, the
frictional resistance to the motion of an individual Brownian (macroscopic) particle has been
supposed to be proportional to the viscosity of the medium, since its velocity is small relative to
stochastic atomic motions. However, this assumption cannot be correct for the diffusion of the
ions themselves, since they move with the same thermal velocity as the atoms of the medium.
Recent computer simulations of glass-forming systems [20, 21] have demonstrated explicitly
a fractional-power relation analogous to equation (4). The relaxation time τα calculated in
reference [20] (which is proportional to η/T ) and the self-diffusion time ∝1/D (proportional
to the reciprocal of ρ/T for a one-component fluid) above the mode-coupling critical point
(MCT) [22] Tc are connected non-linearly:
τα ∝ (1/D)m.
(5)
The estimations in reference [20, 21] of m give values from 0.7 to 0.82, which is very
close to the empirical value m = 0.8 ± 0.1 derived from our measurements [16] for the region
of comparatively high temperature far from the glassification point (see figure 7).
At lower temperature, the corresponding-states law fails to be valid (see figure 4) and the
exponent m has a tendency to decrease near the glassification point (see figure 8, covering a
wider range of viscosity). This behaviour is analogous to that observed by Andreozzi et al
[23] for rotational diffusion in a glassifying liquid. One can see that this m changes differently
for different glass-formers.
Let us try to probe deeper in our attempt to understand the nature of supercooled fluid.
The concept of a sort of heterogeneity in the supercooled system is the key idea of many
recent theories [24–28]. Whatever the reason for the emergence of this heterogeneity, the
corresponding liquid on its nanometric scale can be presented as a composite material with
inclusions of greater rigidity (and, probably, of higher density) [29,30], which live much longer
than the reorientation time of an individual molecule [30].
The conductivity of a composite material consisting of two components (or phases)
of different conductivity (assuming for simplicity the rigid phase to be an insulator—
i.e. its conductivity to be close to zero) can to a first approximation (the ‘effective-medium
approximation’ (EMA) [31]) be presented as
σeff = σ1[1 − p2(T )/pc0 ].
(6)
Here, σeff is a measured quantity, σ1 is the conductivity of the fluid phase, p2(T ) is
the volume fraction of the rigid (insulating) phase, which depends on the temperature. This
dependence is specific to different glass-formers. Linearity of equation (6) is possible until
p2 is small compared to pc0 ; this is connected with a threshold determined by the geometry
of the inner composite structure. For p2 close to the threshold value, the right-hand side of
equation (6) transforms into a complicated function of the inner structure of the material, which
has to be carefully investigated.
For the fluidity ϕ (ϕ = 1/η) of the composite, the analogous formula can be written [32]
as
ϕeff = ϕ1[1 − p2(T )/pc00].
(7)

8782
E Veliyulin et al
Until p2(T ) ꢀ 1, the volume fraction p2 in equation (7) can be considered the same (or,
one hopes, close to the same) as in equation (6), since the clusters which are mechanically
rigid have to be insulating as well. However, pc0 and pc00 for equation (6) and equation (7) may
be essentially different.
One has to remember that though we use the terms ‘phase’ and ‘component’, the above-
mentioned inhomogeneities are not thermodynamically stable. These stochastically emerging
(and disappearing) closer-packed bodies [20, 21, 32, 33] create walls and chains rather than
the compact three-dimensional regions. That is why the limiting figures pc0 and pc00 might be
considerably different from the figures expected from percolation theory [34].
Since one observes for the liquid the universal Arrhenius behaviour in the high-temperature
region of figure 4, where it is actually homogeneous, it is natural to extrapolate this behaviour
for the fluid phase (σ1 and ϕ1) down to the supercooled region in accordance with equation (1)
and its analogy for viscosity.
Thus, until the term p2(T )/pc0 (the collective obstruction effect of the rigid inclusions)
is small compared to 1 (and the index m is not much less than 1), the relation between the
conductivity and fluidity is weakly dependent on the temperature and is determined mostly by
thehigh-temperaturebehaviouroftheliquid(seefigure7). Thedeviationfromuniversalitymay
grow with the growing p2(T )/pc0 term, as is clear from figure 8 where the curves for different
glass-formers deviate from their parallelism at lower temperature. The fact that the exponent
m is less than 1 (so-called ‘decoupling’) is frequently interpreted [30] as a manifestation of
a trend towards glassification. But figure 7 rather convincingly demonstrates an intrinsic
character of the resistivity–viscosity ‘decoupling’ or, perhaps it is better to say, the generality
of the fractional (FSE) relation (4) between them. The trend towards glassification leads to
further reduction of m depending on the individual character of a glass-former determined by
the specific geometry of the clusters which it forms.
Figure 9 presents the deviations of the effective conductivity and fluidity ((σ1 −σeff )/σ1 =
p2(T )/pc0 and (ϕ1 − ϕeff )/ϕ1 = p2(T )/pc00, respectively) from those of the ‘fluid phase’; these
originate from the presence of ‘rigid’ inclusions. In a first approximation, the conductivity
0.5
0.4
0.3
0.2
0.1
0.0
Figure 9. Deviations from the ‘fluid-phase’ conductivity and
fluidity of the effective conductivity (σ1−σeff )/σ1 = p2(T )/pc0
(empty symbols) and fluidity (ϕ1−ϕeff )/ϕ1 = p2(T )/pc00 (filled
500
600
700
800
900
1000
symbols) as functions of temperature. Circles correspond to
T [K]
Ca2Rb3(NO3)7 and squares to ZBLAN20 data.

Universal transport coefficient behaviour in ionic melts
8783
and the viscosity of CRN glass-former give close results. But the stronger decoupling of the
conductivity and viscosity data observed for ZBLAN20 by Hasz et al [7] (see figure 8) reveals
a considerable difference between the individual ionic motions and the collective viscous flow
even within the range of validity of the EMA. The increase of p2(T ) towards the glassification
point probably leads to strong deviations from linearity in both equations (6) and (7). While
for the viscous flow the establishing of an interconnected infinite cluster of the rigid phase
is important (p2 < 0.5), for the conductivity the important limit is the loss of the infinite
cluster of the fluid phase ([1 − p2] < 0.5). The accuracy of phenomenological measurements
is not sufficient to allow a detailed judgment to be made as regards the shape of the rigid
inclusions. But further investigation may produce a meaningful classification of glass-formers
in accordance with the discrepancy between the parameters derived from the conductivity and
viscosity data.
8. Conclusions
Precise resistivity data have been obtained for the ionic pure and mixed melts RbNO3, NaNO3,
Rb3Na2(NO3)5, Ca2K3(NO3)7, Ca2Rb3(NO3)7, Ca3Na1Rb6(NO3)13, (KCl)0.33(AlBr3)0.67
and LiCl–water solutions over a wide range of parameters distant from the glassy state.
These results, together with the data from the literature, allow us to find universal features
in the melt resistivity behaviour which were observed before for viscosity [1,2]. The conduct-
ive behaviour of the melts in the high-temperature region is found to be a universal function
of the scaled temperature (the ‘corresponding-states law’). The activation energy Eρ derived
from the Arrhenius equation has been suggested as a scaling parameter. The parameters for
all of the melts have been tabulated and compared with those derived from viscosity.
In the low-temperature region, which is defined by T /Eρ < 0.2, the universal diffusion
mechanism of transport fails and the effects of the inhomogeneity of the undercooled liquid
are revealed to be important. The Vogel–Fulcher equation, which takes these effects into
account phenomenologically, requires the introduction of an additional characteristic ‘fragility’
parameter, Dρ.
The viscosity–resistivity relation derived from our experimental data appears to be a frac-
tional-power functionwhichisvalidovera muchwiderrange thanthe corresponding-stateslaw.
The preliminary consideration of an intrinsic heterogeneity of two different undercooled
glass-formers is presented.
Acknowledgments
This work was partially supported by the German–Israeli Foundation (GIF) for Science and
Development. This work was also supported by The Aaron Gutwirth Foundation, Allied
Investments Limited (Israel). The authors are grateful to Dr W C Hasz for his kind readiness
to provide us with the experimental data on ZBLAN20 melt.
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