pressure dependence of t in silicate glasses from...
TRANSCRIPT
Pressure dependence of Tg in Sil icate Glasses from
Electrical Impedance Measurements
NIKOLAI S. BAGDASSAROV , JÉRÔME MAUMUS, BRENT POE§ , ANATOLY B. SLUTSKIY* , AND VADIM
K. BULATOV*
Institut für Meteorologie und Geophysik, Universität Frankfurt, Feldbergstraße 47, D-60323Frankfurt/Main, Germany
§Bayerisches Geoinstitut, Universität Bayreuth, Universitätsstraße 30, D-95447 Bayreuth, Germany
*Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, Kosyguine str. 19,GSP-1, 117975 Moscow, Russia
Abstract
The effect of pressure as an external control parameter has been estimated on dielectric relaxation in alkaline (albite),
alkaline earth (anorthite) silicates and silica glasses having differing fragility index. The pressure dependence of the
glass transition temperature (Tg) in silicate glasses correlates with a pressure dependence of the shear viscosity, and,
thus, the measurements of pressure dependence of Tg provide an indirect information of the melt rheology under
pressure. Below Tg activation energy of electrical conductivity is less than that at T> Tg. The intersection point of two
Arrhenius dependencies of electrical conductivity as function of 1/T defines Tg. The electrical impedance
measurements has been performed in atmospheric furnace and in 3 types of high-pressure apparatus: piston-cylinder,
belt and multi-anvil presses. The measurements were conducted in the frequency range 100 kHz - 0.01 Hz using
parallel plate and concentric cylinder geometry of a cell . The measured glass transition temperature in anorthite varies
with pressure P (in GPa): Tg= 850°C + 5°/GPa ·P, in albite glass the pressure dependence on Tg = 685°C - 8°/GPa·P, in
HPG8 Tg = 777°C - 45°/GPa· P, and in sili ca glass Tg =1050°C + 17°/GPa·P. The slope of dTg/dP decreasing with the
pressure increase and the contrast between activation energies of the electrical conductivity below and above Tg is
smeared out. Dielectric relaxation times calculated from the imaginary component of the dielectric modulus are three
orders of magnitude smaller than structural relaxation times at Tg and becomes slower with pressure increase. The
activation energy of dielectric relaxation for sili ca glass above Tg is equal to the activation energy of the structural
relaxation. In albite glass this activation energy corresponds to the activation energy of the Na tracer diffusion. In
sodium bearing glasses Tg estimated from electrical conductivity is a “sodium ion mobilit y” Tg, corresponding to a
overlapped α and β-relaxation processes and therefore, shifted to lower temperatures. With the pressure increase the
activation energy of dielectric relaxation in anorthite increases having the activation volume of ca. 10±5cm³/mol, in
albite glass the activation volume is smaller and negative –2±1 cm³/mol.
Key word: glass transition, pressure, impedance spectroscopy, sili ca, albite, anorthite,
haplogranite
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Introdu ction: Glass transition and Viscosity vs. Pressure
The glass transition temperature Tg of silicates is a homologue temperature at which melts believed
to possess a viscosity c. 1012-13 Pa s and a relaxation time of shear stress c. 100 s [1]. Knowledge of
the glass transition temperature at high pressures provides an indirect information about the
pressure dependent rheology and effect of pressure on relaxation processes. This is of a special
interest in geosciences and in silicate melt physics, where the viscosity measurements at high
pressures and temperatures are technically difficult to carry out: in-situ rheological measurements
need in an implementation of X-ray radiography and synchrotron radiation technique [2].
The question of the pressure dependence of sil icate melt viscosity at high pressures plays a
key role in a prediction of magma accent to the Earth’s surface [3]. For decades earth scientists
have sought a general equation to model viscosity of magmatic sil icate melts as a function of
temperature, pressure and composition (e.g. [4-8]). In early viscosity models [4, 5] the Arrhenian
temperature dependence of viscosity
RT
EA a
T +=ηlog (1)
was suggested, where Ea is the activation energy of viscous flow, A is pre-exponential factor. Eq 1
exploits two parameters both of which depend on melt composition. Recent experimental results
demonstrate that Eq 1 is not adequate for most silicate melts over a wide temperature range [6, 7].
In recent years some new models of viscosity have been applied to silicate melts by the use of VFT
(Vogel-Fulcher-Tammann) equation instead of the Arrhenius type
oT TT
BA
−+=ηlog (2)
[7, 8] or on the basis of the WLF (Will iams-Landel-Ferry) equation
g
gTT TTB
TTAg −+
−−+=
)(loglog ηη (3),
[6]. In Eqs 1-3 ηT is the viscosity at temperature T, A, B, To are the empirical constants, Ea is the
apparent activation energy for viscous flow, ηTg is the viscosity at the glass transition temperature
Tg, and R = 8.314 J mol-1 K-1 is the universal gas constant. In contrast to the temperature
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
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dependence, the pressure dependence of magmatic melt viscosity is poorly constrained. There have
been only a few attempts to generalise the viscosity dependence on pressure: (1) either by
introducing a pressure dependent activation energy in Eq 1 [5], or (2) by applying a WLF-type
equation and „correcting“ the glass transition temperature Tg for pressure
)(
)]([loglog ,1, PTTB
PTTA
g
gTTP g −+
−−+= ηη (4),
[6]. One unclear point in the latter approach concerns the type of glass transition temperature to be
used in Eq 3-4: rheological, calorimetric, dilatometric etc.?
A second unclear point is the dependence of Tg on cooling rate q, which has to be taken into
account by measurung Tg from different probing methods. What method of Tg estimation is relevant
to be used, cooling-heating temperature scans in a time domain or dynamic measurements in a
frequency domain? The general dependence of Tg as a function of cooling rate is as follows
2
ln
gTg RT
G
dT
qd ∆= (5),
where ∆G is the activation energy of a physical property used for detecting Tg [9, 10]. Thus, for
samples with differing thermal histories one may expect differing unrelaxed states of supercooled
melt at Tg and, thus, the uncertainty in viscosity estimations by applying Eq 3-4 may be large. In
any case, the essential part of the systematics between viscosity and pressure may be experimental
or theoretical estimations of a glass transition temperature as a function of pressure for samples
having the same q.
A general way to describe the pressure dependence of Tg(P) is to modify Eq 5 by correcting ∆G at
normal pressure for some pressure-volume effect
∆∆+⋅=
11 1)(
G
VPTPT gg
or
11
G
VT
dP
dTg
g
∆∆⋅= (6),
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where ∆V is the activation volume of a physical property used to determine Tg and index 1 refers to
the normal pressure [11].
A different approach was suggested by considering the glass transition as a kind of second-order
phase transition. In this case, a Clayperon slope of Tg with pressure must obey
p
Tg
dP
dT
αβ
∆∆= (7),
and
P
pg
C
VT
dP
dT
∆∆⋅⋅
=α
(8),
where ∆βT is the jump of the isothermal compressibil ity, ∆αp stands for contrast of isobaric
expansivities, and ∆Cp is the difference in the isobaric heat capacity between liquid and glass [12-
15]. Combination of Eqs 7 and 8 (two Ehrenfest equations) provides a so-called Prigogine-Defay
ratio Π
12
=∆⋅⋅∆⋅∆
=Πpg
Tp
VT
C
αβ
(9).
A number of experimental works have been done on polymers, low temperature glass-formers,
silicates, etc. to test the validity of Eq 7-8 [12, 16-19]. Usually, the value of the PD-ratio is greater
than 1 (Π~ 2÷5). This violation has been explained due to a non-equili brium nature of the glass
transition [20] when one ‘ordering parameter’ does not adequately describe a phase transformation.
In some works this problem was solved by introducing an additional ‘ordering parameter’ to
generalise Ehrenfest equations for the case of the glass transition (e.g. [20; 21]).
In many works on glass transitions it has been demonstrated that the first Ehrenfest equation (Eq 7-
8) may be satisfied automatically by a proper calculation procedure of volume derivatives in
respect to pressure and temperature [13, 14]. In the derivation of the second Eherenfest equation the
free energy must be corrected to the effect of the configurational entropy - Te x Sconf , where Te >T
an effective temperature at which processes related to the configurational entropy are “thermalized”
[14]. In this way the corrected PD-ratio may be very close to 1 for some glass-formers.
Alternatively to the thermodynamic approach mentioned above, there is an attempt to describe the
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pressure dependence of the glass transition by use of the VFT relation (Eq 2) and replacing
temperature by a pressure variable. If τ is the shear stress relaxation time, which relates to the
relaxed shear viscosity as τ ~ η/G (G is unrelaxed shear modulus), then
−
⋅=PP
B
oo expττ at T= const (10),
derived from a free-volume model (e.g. [22]). By analogy with temperature glass transition, the
glass transition pressure characterised by a pressure Pg, at which the relaxation time is 10² s and Po
is an ideal glass transition pressure at which the viscosity exponentially diverges. In some
isothermal experiments this behaviour of viscosity was observed [23-24]. On the basis of the
isothermal dielectric relaxation time measurements on supercooled strong liquids another
expression was suggested
−⋅⋅=
PP
PC
o
po exp,ττ at T= const (11),
[22]. In Eqs 10-11 Po is the pressure of an ideal glass transition (by analogy with To in Eq 2), τo,p
denotes the relaxation time at normal pressure 0.1 MPa. The general expression of relaxation time
(or viscosity) can be considered in this case in the form
−⋅+−
−⋅+⋅=)]([
)(exp
10
1
PPaTT
PPbBoττ (12),
where P1 is atmospheric pressure [25-26]. Eq 12 was derived from Eq 2 by introducing constants B
and To , which linearly depend on pressure. Eq 12 has been tested on specific heat spectroscopic
measurements (enthalpy relaxation) with unsatisfactory results for fragile glass-formers [25]. The
fail of the universal pressure-temperature superposition principle is explained by the fact that the
glass transition is sometimes considered as a simple volume-activated process ( ln τ ~1/Vf and Vf is
free volume). The free volume in this approach is believed to depend on pressure and temperature
in the same way through constant compressibility and thermal expansion coeff icient. This is not
true for many classes of glass-forming materials like organic glass-formers, polymers etc. Contrary
results were obtained for dielectric relaxation in epoxy resin, where relaxation times are suitably
described by the extended VFT-equation, Eq 12 [22]. Finally, to construct an expression for
viscosity of magmatic silicate melts according to Eq 12 we would require 5 fitting parameters
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obtained from a large number of relaxation experiments at different pressures and temperatures.
According to Eq 4 we need only 3 fitting parameters and independent data on Tg(P).
Another important material parameter needed for characterisation of glass-formers is a fragil ity
index, which represents the extent of the liquid’s deviation from Arrhenian behaviour and can be
defined as follows
g
g
TT
gTTg
T
Td
d
TTd
dm
=
=
=
⋅=
)log(
)/(
ln
10ln
1 ητ(13),
(e.g. [27, 28]). From Eq 2 and 13 it follows that the fragil ity index m depends on Tg as
2)(10ln
1
og
g
TT
ATm
−⋅
⋅= (14),
where To <Tg. If Eq 3 is used for the viscosity model, then
B
TAm
g⋅⋅=
10ln
1(14‘).
The behavior of m under pressure is unknown. According to Eq 14 the positive dependence of Tg on
pressure will result in decreasing m (denominator is a stronger function of Tg than nominator), and,
therefore, a liquid under pressure becomes stronger, which was observed in dielectric relaxation
experiments with epoxy resin [22]. If dTg/dP is negative, then m increases with pressure and a
liquid is more fragile. Contrary to conclusions obtained by using Eq 14, the viscosity model
described by Eq 3, predicts an increase or decrease of the fragil ity index m directly proportional to
the pressure dependence of Tg. This simply means, that Eq 3 and 14’ may only be applicable, if the
fragil ity parameter m is small , i.e. for almost strong liquids (Tg>> To).
The relaxation model of Avramov [29] based on the entropy description of the glass
transition predicts viscosity and structural relaxations as a function of pressure and temperature [30,
31]. The model assumes a Poisson distribution of the energy barrier energies of jump frequencies at
temperature of the glass transformation. The temperature and pressure dependence of relaxation
time and fragility index m in the frame of this phenomenological theory is as follows
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
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î
+⋅
⋅⋅=
ba
g
P
P
T
T*0 1exp εττ (15),
where ε∼ 30.5 is the dimensionless activation energy at Tg, a=2⋅Cp/ (Z⋅R) plays a role of a fragile
parameter, Cp is the heat capacity, R is the gas constant, Z is the co-ordination number of the liquid
lattice, b= 2⋅κ0⋅Vm⋅P*/(Z⋅R) is the dimensionless constant, κo is a thermal expansion coeff icient, Vm
is the molar volume, P* is pressure constant estimated from an empirical relationship of the
thermal expansion coefficient upon pressure 1
*0 1−
+⋅=
P
Pκκ [31]. By introducing a pressure
dependence of the co-ordination number Z, Eq 15 demonstrates a good agreement with
experimental results on some polymers glass formers [32]. After substitution of Eq 15 in Eq 13 and
neglecting the pressure dependence of a, b and Cp, the fragility index m depends on pressure as
follows
⋅
+⋅
−⋅
+⋅
== gTT
g
b
dT
dP
PP
Tba
P
P
m)(10ln
1
*
*ε
(16).
Thus, depending on the sign of pressure dependence of the glass transition temperature, the
fragil ity index m may increase or decrease with pressure.
Fig. 1. Angell plot of samples used in the electrical conductivity study. The fragility index varies from 17 for SiO2 to 42 for anorthite[33]. Viscosity data for HPG8 glass are from [34].
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
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The main purpose of this paper is to demonstrate possibilities and limitations of Tg and the pressure
derivative dTg/dP measurements obtained from dielectric spectroscopy at high pressures on sil icate
glasses having differing fragility index m.
Experiments
Description of Glass Samples
Compositions of sil icate glass samples that were used in impedance spectroscopy experiments are
listed in Table 1. In the present study we have compared the pressure dependence of 4 different
glasses. Their fragil ity index m calculated according to Eq 13 is shown in Fig. 1. Besides the
differing fragil ity index m and plots of viscosity vs. Tg/T in a spirit of the Angell classification (e. g.
[33], these glasses have very different structures and mechanisms of the electrical conductivity.
They can be classified as “ loose” (albite and haplo-granitic glasses), “tight” and “fragile”
(anorthite), and “tight” and “strong” (SiO2) conducting glasses [35], accordingly. The relationship
between dispersion, polarisation, glass transition and “mobile ion Tg“ of these glasses will be
discussed below.
Table 1. Composition of glass samples (in wt. %)
CaAl2Si2O8 NaAlSi3O8Oxide
Startingcomposition
Afterexperiment
Startingcomposition
Afterexperiment
HPG8
SiO2 42.63 42.62 67.76 67.53 77.9
Al2O3 36.86 37.14 19.58 21.37 11.89
Na2O 12.09 10.39 4.53
K2O 0.21 0.27 4.17
CaO 20.13 19.89 0.18 0.21
Sum: 99.62 99.65 99.82 99.77 98.48
Cooling rate, K/h 20 20 300
Microprobe analysis JEOL JXA – 8900RL, 20 kV, sample current 20nA, average of 3-5 points
The samples of anorthite and albite glass were taken from the collection of silicate glasses used by
Prof. M. Rosenhauer (Göttingen), the sample of HPG8 was given by Prof. D. Dingwell (München),
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
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and the sample of SiO2 glass is a commercial glass Suprasil 300Ò (Heraeus, Germany), containing
< 1 ppm of OH-. The synthetic haplo-granitic composition (HPG8) corresponds to the eutectic
composition of SiO2-NaAlSi3O8-KAlSi3O8 at 0.1 GPa PH2O and models well a calcium-free granite
system. The interest of using the HPG8 sample in the present experiments is that many physical
properties of this glass have been measured in recent years [36]. The difference in physical
properties of this glass and albite may be explained as a mix-alkaline effect.
Estimation o f Tg
The electrical conductivity was used as a probing tool to detect Tg at high pressures. The impedance
measurements were performed on glass samples of anorthite (CaAl2Si2O8), albite (NaAlSi3O8),
SiO2, a haplo-granite (HPG8) composition and SiO2. To estimate a glass transition temperature at
pressure we have used a method of defining the activation energy of bulk electrical impedance (or
conductivity) in the glass transition temperature range. Despite a “decoupling” of the macroscopic
viscosity and its relaxation time from electrical conductivity relaxation, the kink in a slope of the
bulk electrical conductivity plotted as function of 1/T can be used as an indicator of Tg [37]. The
experiments have been done on silica, albite, anorthite and haplogranite glasses in an atmospheric
furnace at 0.1 MPa, in a piston-cylinder apparatus at pressures from 0.3 to 1 GPa, in a belt-
apparatus between 3 - 4.5 GPa, and in a multi-anvil press at 6 GPa.
Electrical impedance measurements were carried out using a Solartron 1260 Phase-Gain-
Analyser interfaced with a PC. The device permits a single sine drive and analysis of a system
under test over the frequency range 10µHz to 32 MHz. In the high-pressure experiments we applied
a 1 V sine signal over the frequency range 0.01 Hz to 100 kHz. Typically, the frequency scan
utilised logarithmic steps of 0.2 - 0.5. Signals at higher frequencies were affected by cable
impedance, and at lower frequency signals became too noisy. The estimation of the “bulk”
electrical conductivity from frequency scans provide a “true” dc conductivity data, and, therefore,
the artefact due to a mismatch of the probe frequency and material relaxation times is solved [37].
Measurements at 0.1 MPa
For measurements in the atmospheric furnace (Fig. 2), glass samples (diameter D = 6 mm, 8 mm
for silica glass) were drilled out of blocks of glass and cut in discs with thickness L = 1 - 1.5 mm.
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Flat surfaces were polished with 0.05µ alumina powder under water up to optical quality of the
surface. The sample of sil ica glass was polished down to 0.5 mm thickness. Pt electrodes (thickness
= 5µ) were sputtered onto both flat surfaces of the glass discs (spot diameter 4.85 mm). On the
silica sample the electrodes were sputtered with a spot of 8 mm in diameter. The samples were
mounted in the inductive furnace (Heraeus , Hanau, Germany) with the heating element made of
Pt-wire. During the measurements samples were fixed between alumina rods (Frialit-Degussit
with flat electrodes made from Pt-foil. The elongation of the sample was measured with a
micrometer gauge having a precision of 0.001 mm. Electrical contact between Pt-foil and sputtered
electrodes was provided by a light flat spring acting axially through one of the alumina rods. The
heating rate of the furnace was approximately 20°/h. A type S thermocouple (Pt-PtRh10) touched
the Pt-foil which was in direct contact with the sample. Electrical impedance measurements were
done only once on each sample during heating cycle. The geometric factor of samples G= πD²/4L,
for albite, anorthite and HPG8 ranged from 1 to 1.3 cm, and for silica glass was approximately 10
cm. Temperature was measured and controlled using a Eurotherm 818P.
Fig. 2. Scheme of the electrical conductivity experiments at 0.1 MPa: 1 – Pt-wire furnace ROR Heraeus , 2 – Al2O3
ceramic rods AL23 Frialit-Degussit , 3 – insulating casting body; 4 – inlet for inert gas (Ar), 5 – flat spring, 6 –electrode contact of Pt-wire, 7 –thermocouple S-type Pt-PtRh, 8 – sample of glass, 9 – sputtered area of sample, 10 –tephlon guiding plug, 11 – Pt-folio cemented to the surface of Al2O3 rods, 12 – inner Al2O3 tube, 13 – elongationgauge.
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Measurements in piston cylinder apparatus
For the moderate pressure experiments (0.3-1 GPa) we used a conventional piston-cylinder
apparatus with an end load. The measurements of the electrical impedance were performed at
pressures up to 1 GPa and temperatures up to 1200°C. The press (max. load ca. 220 tonnes)
consists of 2 independent hydraulic cylinders: the first provides an end-load and the second moves
the piston at the sample assembly. The diameter of the high-pressure autoclave is ½” (Danfoss ,
Denmark). The inner part of a high-pressure cell is shown in Fig. 3. The pressure calibration of the
cell was determined using some standard point materials: at room temperature the transformations
Bi I-II- III at 2.56 GPa and 2.7 GPa [38] were used; at high temperature, melting curves of NaCl
and CsCl [39] and the α−β transition in LiNaSO4 [40] were used as standard points. Melting points
of NaCl and CsCl as a function of pressure up to 2.5 GPa has been determined in-situ by electrical
conductivity measurements. At a melting point the observed drop in the electrical impedance at 1
kHz was ca. 2 orders of magnitude. The pressure calibration is believed to be within an accuracy of
± 0.03 GPa. The temperature gradient in the cell has been estimated on dummy samples of pressed
Al2O3 powder by monitoring three separate thermocouples. We estimate a radial temperature
gradient of ca. 1°/mm, and a vertical temperature gradient of ca. 2°/mm in the temperature range up
to 900°C.
A cell for electrical impedance measurements in the piston cylinder utilised a coaxial cylindrical
capacitor with a geometric factor of 7-8 cm filled with the sample under test. The exact geometric
factor of the cell was evaluated independently from calibration measurements on NaCl solutions
(0.01M - 3 M) at 22°C and atmospheric pressure. For these purposes, a cylindrical gap between
two Pt-electrodes (made of two Pt-tube 0.1 mm in thickness with outer diameters 2.2 and 4 mm)
has been filled with a NaCl-solution of a known molar concentration. The measured conductivity of
NaCl-solutions was compared with table values [41]. The difference between the calculated
geometric factor of a cylindrical capacitor
=
d
D
LG
ln2π (17),
and the measured geometric factor was about 25%, where D is the diameter of an outer electrode, d
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is the diameter of an inner electrode, L is the length of the cylinder. Measurements of the geometric
factor of the cell after high-pressure experiments revealed that due to the cell deformation under
pressure L increases by ~3-4%, D increases by 1-2%, and d remains constant. Overall , these
variations of the cell dimensions did not affect the results of Tg or dielectric relaxation
measurements.
During impedance measurements in the piston-cylinder, the mass of press was isolated from the
ground of the Solartron 1260. Wires from the Pt-thermocouple and the mass of the high-pressure
autoclave were used to connect the measuring device and the cell electrodes. Before doing the
high-pressure experiments a measuring cell was calibrated for short-circuit and open-circuit
impedances over the frequency range 1 MHz - 0.01 Hz. A typical AC-resistance of the cell to a
short connection was 0.4 Ω. These calibrations have been taken into account in the final
calculations of the electrical impedance as a function of frequency. At high pressure and
temperature the measurements of the electrical impedance were conducted without an automatic
temperature control in order to reduce the electrical noise produced from the heater regulation.
During electrical conductivity measurements, the Eurotherm 818P controller was switched off,
and the temperature was regulated manually through a variable transformer.
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Fig. 3. Principal scheme of a piston-cylinder cell used for electrical impedance measurements [40]: 1 - inner cylindricalPt-electrode; 2 - outer cylindrical Pt-electrode; 3 - sample; 4 - thermocouple S-type; 5 - Al2O3 -ceramic Alsint-99.7; 6 -graphite heater; 7 - CaF2 pressure transmitting medium; 8 - bornitride; 9 – boron nitride ceramic; 10 - hard metal core;11 - hard metal piston; 12 - plug of stainless steel; 13 - unfired pyrophyllit e; 14 - copper-ring; 15 - ground.
Experiments in belt apparatus and multianvil press
A detailed description of the belt-apparatus can be found elsewhere [42]. The construction of the
experimental cells for belt- (Fig. 4) and multi-anvil apparatus (Fig. 5) has the same principle used
by Xu et al. [43]. The geometry of the cell was a parallel-plate capacitor with a geometric factor
approximately 0.3 cm. Metallic electrodes were made from Mo-foil . The cell is protected from
electrical noise of the graphite (belt-apparatus) or LaCrO3 (multi-anvil press) heater by a grounded
shield of Mo-foil. During the electrical impedance measurements in both apparatus the automatic
temperature control was switched off . The temperature was regulated manually through a power
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thyristor.
Fig. 4. Principal scheme of the electrical impedance measurement in belt apparatus (height 16 mm [42]). 1 – sample(diameter 2.1 mm, thickness 1 mm); 2 – electrical shield from Pt-foil; 3 – electrodes from Pt-foil ; 4 – Al2O3 ceramics;5 - thermocouple; 6 – pyrophyllit e 7 - graphite heater; 8 – boron nitride.
On each sample the electrical conductivity was measured only during a single heating cycle. In
both configurations, one of the thermocouple wires was connected to one electrode of the cell with
the AC-bridge. During the frequency scan the temperature indicator was disconnected from the
thermocouple in order to avoid coupling of the two devices at low frequencies.
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Fig. 5. A. Principal scheme of electrical conductivity measurements in multi-anvil press [43]; B. The cross-section ofthe cell with anorthite glass sample after experiment. Thickness of the sample is 1 mm, diameter 2.1 mm.
For each glass sample, we measured the impedance over the frequency range 0.01 to 105 Hz. A
bulk electrical conductivity of glasses has been estimated from Argand plots generated from these
measurements or by fitting procedure of impedance spectra discussed below. Fig. 6 ill ustrate the
Argand plots measure for SiO2 at 0.5 GPa. The intersection of the arc with the Re(Z) axes defines
the bulk resistance of the sample. In some cases we compared these electrical conductivity
measurements with an electrical conductivity measured at a constant frequency 10³ Hz. Arrhenius
plots of the electrical measurements of the bulk resistance were measured covering a temperature
range both below and above the glass transition. The plots indicate a change in slope of the
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conductivity curve at the glass temperature Tg. Measurements from the two methods (bulk
conductivity measurement from Argand plot and isofrequency measurement) provide the same Tg
for both albite and haplogranite glasses but result in significantly differing Tg values for both
anorthite and silica glasses. The reason for the difference between fixed frequency measurements
from a dc conductivity curve and systematically lower values of activation energies obtained from
this method has been discussed in [37].
Fig. 6. Argand diagram of Suprasil 300 sili ca glass. The bulk resistance is estimated from intersection Im(Z)-plot withthe Re(Z)-axes. The impedance arc represents a perfect semicircle. Thus, sili ca glass belongs to a class of “ tight” andstrong glassy conductors [35].
Results
Results of 0.1 MPa experiments
The results of electrical conductivity measurements at 0.1 MPa on albite, haplo-granitic, anorthite
and sil ica glasses are presented in Fig. 7. The bulk conductivity σdc has been estimated from
Argand-plots as an intersection of –Im(Z) graph with Re(Z)-axes [48]. On an Arrhenius type of
diagram the σdc data show two distinct slopes with differing activation energies:
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−⋅=
RT
Eadcdc exp,0σσ (18).
The kink-point on plots ln(σdc) vs 1/T may be used as a characteristic temperature point to
discriminate glassy and liquid states (e.g. [44, 49]).
Electrical conductivities of Na-bearing glasses (haplogranite, albite) are 104 to 105 times higher
than those of anorthite and SiO2 glasses at comparable temperatures. This difference is attributed to
the easy dipole polarisation of sodium bearing glasses. High conductivity in these glasses
associated with mobil ity of Na+ ions which diffusivity is higher than other species. Above Tg the
rotation and translation movement of Na-cations around non-bridging oxygen atoms contribute to
the electrical conductivity of Na+-bearing glasses at high frequencies (> 10² Hz). The observed
kink of the slope in an Arrhenius plot ln|Z| vs. 1/T (where Z is the bulk impedance and T is the
absolute temperature) measured at high pressure for Na-bearing glasses is significantly lower (>
50°C) than Tg estimated from calorimetric and rheological measurements at 0.1 MPa [47, 50]. For
anorthite and silica glasses the observed kink in the slope is slightly lower than the calorimetric Tg
[51].
For glasses with alkalis, like Na+ in albite and haplo-granitic glass, the activation energy of σdc
depends on the mobil ity of Na+ in the structure. On cooling, at a certain temperature the mobility of
a translation motion of alkali ions is not correlated with the thermal structural modifications. In the
melt phase, structural elements may always be arranged in an energetically favourable network,
leaving minimum space for mobile charged species like Na+. Thus, the activation energy of σdc
above the glass transition temperature may be higher than below. Thus, for alkali -bearing systems
the temperature at which the kink point is observed in electrical conductivity measurementsis
different from the glass transition temperature estimated from rheological, heat capacity, or
dilatometric data (see Table 2). The measured activation energies of σdc for albite differs strongly at
T < 683°C (Ea= 55 kJ/mol), and at T > 683°C (Ea = 90 kJ/mol). For HPG8 at T < 774°C, Ea = 90
kJ/mol, whereas at T > 774°C, Ea= 120 kJ/mol. For anorthite at T < 844°C, Ea= 42 kJ/mol, and at T
> 744°C Ea= 80 kJ/mol. For silica glass at T < 1050°C, Ea = 100 kJ/mol, and at T > 1050°C Ea=
260 kJ/mol. In previous estimations of Ea from electrical measurements above and below Tg the
reported Ea values are significantly different: for albite Ea is ~57 and ~60 kJ/mol below and above
Tg, respectively [44]; for anorthite at 10 kHz Ea is 118 and 9.3 kJ/mol [49].
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
18
For silica glass containing OH-impurities the reported value of Ea for σdc at T < Tg is ~ 97 kJ/mol
[46]. Measurements on silica glass films at temperatures well below Tg (20-300°C) indicate a
much smaller activation energy for σdc ~39 kJ/mol [61] identified as being the hole-like polaron
hopping energy.
Fig. 7. Results of the bulk electrical resistance σdc measurements at 0.1 MPa. The kink in the slope of the Arreniusdependence of the bulk electrical conductivity indicates the glass transition temperature for mobile charge carriers(cations) in the structure of Ab, HPG and An glasses. In SiO2 glass the charge carriers are impurities, li ke OH-speciesand alkalis.
Structure of glasses and electrical cond uctivity at 0.1 MPa
The significant difference in electrical conductivity of sil ica and feldspar glasses might be
understood from their differing structures. The structure of feldspar and silica glasses has been
determined by X-ray radial distribution analysis [62]. Silica glass has a trydimite-like bonding
topology based on stuffed six-membered rings of SiO4 tetrahedra. Albite glass has the same
structure as silica with aluminium substituting for silicon in some of the tetrahedral sites. In order
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
19
to accommodate sodium cations such a structure has more „void spaces and interstices“ between
rings than SiO2 [62]. In anorthite the glass structure is based on four-membered rings of SiO4 and
AlO4 tetrahedra.
The mechanism of electrical conductivity in feldspar and silica glasses may be described as a
hopping of alkali-ion (Na+ in albite; Na+ and K+ in HPG8) or Ca2+ (in anorthite) from one non-
bridging oxygen site (NBO) to another. In sili ca glass the electrical conductivity mechanism is a
hopping process of OH- and alkalis impurities between defect [AlO4-M]0 tetrahedra (AlO4-alkalie
metal centres), as well as interconversion of charged NBO sites by hopping of polarons and
electrons [47, 63]. The relative effectiveness of a charge transport mechanism in glasses will
depend on average distance and value of energetic barriers between neighbouring NBO-sites or
defected SiO4-tetraedra (hopping activation energy). It will depend also on the concentration of
NBO-sites and impurities, and on the relative size of hopping electric charge carriers (Na+, K+, Ca2+
or OH-). The highest electrical conductivity was observed in albite glass rather than in HPG8. In
HPG8 glass Na-ions are partially substituted by larger K-ions, and the concentration of alkali i ons
is also smaller than in albite (Table 1). The least conductive of the feldspar glasses is anorthite. In
anorthite glass the structure is less open compared to albite. Average interatomic distances, which
might be correlated to hopping distances, are 1.63 and 1.66 Å for T-O, and 3.12 and 3.15 Å for T-T
for albite and anorthite, respectively [62]. In anorthite Al and Si are the most ordered in comparison
with other feldspar glasses, i.e. substitution of Al to Si is less random than in alkali
aluminosil icates. Sili ca glass, Suprasil 300, used in this study possesses the lowest electrical
conductivity due to much smaller concentration of defects, OH-groups and [AlO4-M]0 tetrahedra.
The lower values of σdc for SiO2 and anorthite glasses in comparison with albite and haplo-granitic
glasses is due to the greater distances between NBO sites and other Si+ and O- defects in the glass
structure. Effectively, the longer distance for hopping electrical carriers results in fewer percolation
paths for conduction [63].
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
20
Table 2 . Glass transition temperature Tg (° C) at 0.1 MPa obtained by different
methods
Glass Cooling
rate, K/h
Viscosity Calorimetry &
DTA
Volume expansion Electrical
conductivity
Light
scattering
SiO2 slow(?)
slow(?)
100
1150 [54] 1180 [52]
1090 [55]
1100 [53] 1050 [t. s.] 1100 [18]
1050 [56]
CaAl2Si2O6 20
20
slow (?)
12
40
600
845 [57]
878 [59]
875 [59]
836 [48]
813 [58]
820 [49]
844 [t. s.]
NaAlSi3O8 20
slow(?)
20
12
40
765 [57]
763 [58]
737 [59]
734 [59]
678-687 [49]
683 [t. s.]
HPG8 300 864 [H96] 856 [34]
800 [60]
774 [t. s.]
[t. s.] – this study
Experiments in p iston-cylinder
For high pressure experiments in piston-cylinder samples of glasses were ground into powder and
pressed into the gap between two coaxial electrodes (see Fig. 3). The bulk electrical conductivity
was measured with a temperature step ca. 10°, and plotted on an Argand diagram. The electrical
impedance data were collected in a scan with 0.17 log step of frequency between 105 and 10-1 Hz.
Imaginary component of Z for anorthite glass in the glass transition temperature range is shown in
Fig. 8. The data of bulk resistance of anorthite presented on an Arrhenius plot demonstrate a
significant change of the activation energy of Ζdc of anorthite glass below and above Tg (see Fig.9)
allowing the determination of Tg with an accuracy ±5°. In albite and HPG8 samples there is a
noticeable frequency dependence of Z at low frequencies (Fig. 10). Plotted in double log co-
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
21
ordinates, the frequency dependence of Z allows estimation of a maximum frequency, F, at which
the polarisation is observed at each temperature. The temperature dependence of the occurrence of
this frequency F differs above and below Tg, providing an alternative way to estimate the glass
transition temperature, as shown in Fig. 11. The albite glass possesses a rather high electrical
conductivity, and, therefore, may be classified as a “ loose” conductor. This class of conducting
glasses has also high polarisation (polarisation exponent is <<1) and a broad dispersion range of
dielectric properties [35]. The estimation of Tg in albite at 0.5 and 1 GPa are shown in Fig. 12.
Sili ca glass is probably a strong and “tight” glass having low electrical conductivity and
polarisation exponent close to 1. In the Argand diagram the bulk properties of SiO2 glass are
marked by a perfect semi-circle (Fig. 5). Estimation of Tg at 0.6 GPa in Suprasil 300 is shown in
Fig. 13. The difference in activation energies of σdc for SiO2 and albite are very small resulting in
an error of Tg ±10°C (Fig.12-13).
Dielectr ic Relaxation Peak in Anorthite at 0.3 GPa
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05
f, Hz
Z"p
, ohm
s
1013°C exp. 1013°C fitting (n =0.90)994°C 994°C (0.93)960°C 960°C (0.94)926°C 926°C (0.95)900°C 900°C (0.97)856°C 856°C (0.99)833°C 833°C (1)814°C 814°C (1)800°C 800°C (1)
dielectr ic relaxation peak
Fig. 8. The imaginary component of electrical impedance of anorthite at 0.3 GPa. The frequency scans have been fittedto Eq 19 for each temperature. With the temperature increase the dielectric relaxation peak shifts toward highfrequencies as indicated by the arrow. Numbers in brackets are estimated values of n. At T>Tg n becomes less than 1.
For albite, similar to the behavior of viscosity vs. pressure, Tg also decreases with increasing
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
22
pressure. The same effect of pressure is observed for the haplogranitic glass. For anorthite and
silica glasses Tg is an increasing function of pressure, as may be the viscosity of these melts.
The data of electrical impedance Im(Zp), where Zp = Zp’ – j·Z’’ p. for anorthite glass at 0.3 GPa are
shown in Fig. 8. The data can be fitted to a sum of two relaxation functions:
nld
ppe
pj
R
j
RZ
)(1)(1 ..
2
..
1
τωτω ⋅⋅++
⋅⋅+= (19),
where the first term corresponds to the electrode polarisation process and the second term stands
for dielectric losses in the sample [44]. The results of fitting are shown in Fig. 8. The temperature
dependence of d.l. determines the activation energy of the dielectric loss peak c.110 kJ/mol. The
parameter n in Eq 19 characterises the broadness of the dielectric loss peak in comparison with a
Debye peak (n=1). As it follows from Fig. 8 by crossing the glass transition temperature m
becomes less than 1 (n ~0.95). Thus, the anorthite is fragile and “tight” glassy conductor [35].
Fig. 9. Results of electrical resistance measurements in anorthite glass at 0.3 and 0.6 GPa. At 0.3 GPa measurements ata fixed frequency 10³ Hz are compared with bulk resistance obtained from Argand plots. Bulk resistance measurementsprovide higher values of Tg (solid arrows) in comparison with measurements at fixed frequency (dotted arrows).
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
23
Fig. 10. Method of estimation of the polarisation frequency F. The intersection point of two derivativesdlog(Z’ )/dlog(f) calculated at f>>1 Hz and f<<1 Hz, corresponds to “mobile Na-ion Tgs” .
Fig. 11. Estimations of glass transition temperature from electrical conductivity measurements in HPG8 glass at 0.3(solid arrow) and 0.6 GPa (dotted arrow) and fixed frequency 10³ Hz. At 0.6 GPa the data of polarisation frequency F(Hz) are plotted as a function of 1/T,K. The kink in the slope on the F-graph corresponds to the kink in the slope on theelectrical impedance curve.
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
24
Fig. 12. Glass transition temperature in albite glass under pressure 0.5 and 1 GPa. Arrows indicate Tg, the temperature
point at which the activation energy starts to increase.
Table 3. Results of Tg (° C) from electrical conductivity data
Pressure, GPa Albite Anorthite HPG8 Suprasil 300
0.1 MPa 683±2 844±3 774±3 1050±5
0.3 GPa 850±3 769±2
0.5 690±3
0.6 863±3 747±3 1060±5
1 675±5
3 662±5 863±5
4.5 871±3
6 635±10 880±10
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
25
Fig. 13. Estimation of the glass transition temperature in Suprasil 300 at 0.6 GPa from the change in the activation energy of thebulk resistance from impedance frequency scans and at fixed frequency 10³ Hz. The Argand plots are shown in Fig. 6.
Fig. 14. Results of Tg measurements in the belt apparatus on anorthite and albite glasses.
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
26
Experiments in Belt-Apparatus and Multi-Anvil Press
The results of electrical conductivity measurements carried out in the belt apparatus are shown in
Fig. 14. The change in the slope of the bulk conductivity below and above Tg is very small for
anorthite as a result of increasing pressure and pressure-dependent compressibil ity of the glass. In
albite glass the difference in the activation energy below and above Tg is significant. In reality, the
kink of the slope in the dependence log(σ) vs. 1/T,K for albite glass corresponds to “mobile Na+
Tgs” because of the presence of “ loose” species Na+ in the structure [35]. Data [64] for albite glass
at T<Tg and pressures 3-4 GPa are in a good agreement with this study. With the pressure increase
the free volume becomes so small that above and below Tg the difference of activation energy of
hopping process of electrically charged defects or defect-sites becomes rather small [64, 65].
Fig. 15. Estimation of glass transition temperature under 6 GPa in multi-anvil press in albite (A) and anorthite (B).
Only two experiments were done in the multi-anvil press on anorthite and albite glasses. Tg in both
samples was very hard to detect, but likely corresponds to a maximum decrease of the bulk
resistance on the Arhhenius plot (Fig 15).
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
27
Discussion of Results
Fig. 16 represents a pressure dependence of measured Tg from electrical impedance measurements
in four type of experimental facil ities: atmospheric furnace, piston-cylinder, belt- and multi-anvil
presses. Table 3 summarises the glass transition temperature estimated from electrical conductivity
measurements in this study. The absolute value of the derivative dTg/dP decreases with the pressure
increase, however, as noted above, Tg is diff icult to estimate at high pressures due to the pressure
dependent compressibility. To detect a correct change in glass temperature with pressure at P = 0.1
MPa perhaps only a short pressure interval is necessary (< 0.1 GPa). Theoretically, the distribution
of relaxation times will broaden at high pressure [66] and the transition occurs over a wider
temperature range. This would make it impossible to detect Tg at some critical pressure. In the
method which was used in this study, this happens when the activation energies of electrical
conductivity below and above Tg become equal.
Table 4 lists previous estimations of the pressure dependence of Tg. Only one experimental work
[67] reports results of in-situ high pressure dilatometric measurements of Tg on glass samples
having very fast cooling rates. dTg/dP measured for diopside glass is in a good agreement with
calculated glass transition slope from the second Ehrenfest equation [47, 68] and viscosity
measurements [6]. The direct measurements of dTg/dP on other glass formers indicate that the
pressure dependence of Tg may vary from 200- 150 K/GPa for B2O3 or lead [69] to 3.6 K/GPa for
metallic glasses [19].
Table 4. Pressure Dependence of Glass Transition Temperature of Some Sili cate
Glasses dTg/dP (K/GPa)
Cooling rate Diopside Albite Anorthite
[67] 22-28 °C/min 37±3 -25±10
[6] calculated from
viscosity
34±14 -70±8 -4±17
[68] 49±3 108±40
Calculated from Eq 6 and ∆V* and Ea ofdielectric relaxation in this study
20°/h -20±100 100±100
from electrical conductivity of this study 20°/h -8±3 +5±5
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
28
1 DTA measurements at pressures up to 0.7 GPa, ² indirect estimations using WLF-equation at 0.1 MPa and the
viscosity data at pressures up to 1.5 GPa (no cooling rate are reported), 3 calculated from thermodynamic data and Eq 8,
not cooling rate reported. In the original paper this result is swapped with the result of [67].
The rheological measurements of albite melt has been studied under pressure up to 7 [2, 3]. It was
established that the decrease of albite melt viscosity is about 0.38 log[Pa s]/GPa. Using temperature
dependence of viscosity of albite melt, the estimation of the glass transition temperature in albite
glass is c. 58 K/GPa.
The glass transition temperature in sil ica glass is the most poorly constrained. Tg may vary from
>1200°C according rheological data [70] and calorimetry [52]) to 1100-1050°C according to high-
temperature light scattering experiments [18, 56] depending on cooling rate, content of OH-, and
annealing temperature. The Prigogin-Defay ratio for SiO2 glass Π ~ 2·105 (!) is unexpectedly high,
indicating that a single relaxation parameter to describe a glassy state is not [18].
Fig. 16. Pressure dependence of the glass transition temperature from electrical impedance measurements. The slope ofthe pressure dependence of Tg at low pressures is higher than in the whole pressure range.
In order to estimate a dielectric relaxation time from measured electrical impedance data (Z* ) the
expressions for the complex modulus as follows were used [44]:
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
29
GZiMiM 0* εω=′′−′ (20),
where G is a geometric factor, ε0 is the dielectric constant of a free space, ω is angular frequency.
The dielectric relaxation times were calculated from frequency corresponding to a peak of M“
1)2( −= peakfπτ (21).
The alternative method to estimate a dielectric relaxation time is to make a fitting of the electrical
impedance spectrum to an equivalent electric circuit, for example Eq 19. As a result of fitting of the n, polarisation exponent [65].
Both methods Eqs 19 and 21 must give the same result, if the impedance data only in the frequency
band around dielectric relaxation peak are used. The activation energy of ! DC may depend on" # $ % # & ' " ( & # ) * " + # , ' $ # ' , - .
RC , where R resistivity and C capacity at a peak frequency, only
when R and C have a same activation energies. The tracer diffusion coeff icient is inversely/ 0 1 / 1 0 2 3 1 4 5 6 2 1 2 7 8 7 1 / / 3 4 9 2 3 : 8 ; < = > ? > @ A B C @ D ? ? E @ F D B G > H H > = > @ A I J D K H J A F L I J M L I A F D N > A F > I D M >D @ A C N D A C < K > K > B G E D I A F > H C > ? > @ A B C @ B > ? D O D A C < K P Q R S T U V T W X S Y X Z Z S Y [ V Z \ S T ] ^ [ V W \ S \ _ W ` W T Y ] \ aEinstein equation the activation energy of the ion trace diffusion ED and Ea of b c d DC must have the
same values.
Fig. 17. Temperature dependencies of dielectric vs. structural relaxation time for sili cate glasses. Dielectric relaxation
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
30
time has been estimated from impedance measurements by using Eq. 21. The structural relaxation time has beencalculated from shear viscosity data at 0.1 MPa by using the Maxwell relationship τ (s) = η (Pa s) /25 GPa. For SiO2
(solid bold line) the data are from [46]; for anorthite (solid thin line) the shear viscosity data are from [47, 71, 72]; foralbite (dashed line): 1 – shear viscosity from [71, 73], 2 – from [47, 68, 74]; for haplo-granitic glass HPG8 (dotted line)the viscosity data are from [34, 75].
The results of dielectric relaxation measurements for silicate glasses are presented in Fig. 17, and
the calculated activation energies of dielectric relaxation are listed in table 4 and compared with the
tracer diffusion activation energies. The values of dielectric relaxation times are much smaller than
structural or shear stress relaxation for about 4 orders of magnitude.
For SiO2 the activation energy of dielectric relaxation below Tg corresponds to tracer diffusion of Si
ca. 220 kJ/mol and secondary relaxation. Above the glass transition Ea of dielectric relaxation is
close to Ea of O-trace diffusion or structural relaxation ~515 kJ/mol [46].
For anorthite, the reported activation energies are for chemical diffusion ~ 230-460 kJ/mol and
significantly larger than measured Ea of dielectric relaxation 75 – 125 kJ/mol. The compiled data
on tracer diffusion in the CaO-Al2O3-SiO2 (40-20-40 wt. %) system indicate that in the glass (600-
900°C) Ea ~ 245 (for 18O) and ~ 255 (for 30Si) kJ/mol [76]. In the melt (1550-1350°C) Ea ~ 230 (for26Al), 290 (for 31Si) and ~ 380 (for 18O). In the supercooled liquid regime at temperatures just
above Tg the activation energy for 18O tracer diffusion may even be 900 kJ/mol. Smaller activation
energies for tracer diffusion of Ca, Al, Si, and O were obtained from molecular dynamic
simulations (MD), for anorthite glass (~ 110 kJ/mol), for melt (~ 180 kJ/mol) [77]. The activation
energy of dielectric relaxation in albite correlates well with the tracer diffusion of Na and differs
from those of Si, Al, and O. From MD simulations, the tracer diffusion coeff icients in [78] have
activation energies ~80 kJ/mol for Na, ~280 kJ/mol for Si, ~ 290 KJ/mol for O, and ~270 kJ/mol
for Al. In general, Ca2+ and Al3+ in sil icate melts have rather low mobilities and their presence in
the alkaline aluminosilicates may even impede the motion of alkali ions [79]. Thus, the estimated
Tg in albite glass from electrical impedance measurements characterises only a hopping and charge
separation relaxation process of sodium and not related to the structural relaxation. The same is
probably true for HPG8. The small scale hopping and charge separation in the electric field is an
effective mechanism of the electrical conductivity in sodium aluminosil icates [80]. The hopping of
sodium ions between positions adjacent to NBO- and Al-tetrahedra sites may explain the high
electrical conductivity of these glasses. The lower values of Tg estimated for albite and HPG8 from
electrical impedance measurements in comparison with calorimetric and dilatometric data are
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
31
indicators of the beginning of a “loose” mobil ity of Na+ in the structure at temperatures below
rheological or calorimetric Tg (Table 2). The idea of a “decoupling” character of dielectric
relaxation from mechanical spectroscopy relaxation in sodium-bearing glasses was discussed in
[81] and, subsequently, was compiled and adapted for geologists in [82]. Activation energies of
dielectric relaxation measured in this study are listed in Table 5 and compared with diffusion
activation energy of cations.
Table 5. Activation energy of diffusion and d ielectric relaxation D.R. (in kJ /mol)
Ions SiO2 CalAl2Si2O8 NaAlSi3O8 KAlSi3O8
OH- 83 (1100-500°C)1
Si 237 (1200-800)2
579 (1410-1113) 3
230 (845-765) 6 337 (1500-1100°C)8
O 48 (250°C)4
516 (2500-1250°C)5
247 (760-760) 6
Ca 327(1600-1350)7
Al 462 (1600-1350)7 425 (1500-1100°C)8
Na 56 (800-350)9 79 (800-350) 9
K 99.5 (1000-350) 9 63 (1000-350)9
D.R.10
this
study.
Eε
222 (0.1 MPa, <Tg)
515 (0.1 MPa >Tg)
226 (0.6 MPa)
74 (0.1 MPa)
111 (0.3 GPa)
138 (4.5 GPa)
43 (0.1 MPa, <Tg)
66 (0.1 MPa, >Tg)
65 (1 GPa)
64 (0.1 MPa)11
98 (0.1 MPa, HPG8)
40 (0.1MPa) 11
1 effective OH diffusion [83], 2 effective SiOSi diffusion [83], 3 tracer diffusion 30Si [84], 4 18O diffusion in sili ca glass
[85], 5 estimated from viscosity measurements and Eyring equation [86], 6 tracer diffusion of 18O and 30Si in CaAlSiO4.5
[84], 7 chemical diffusion of CaO/MgO and Al2O3 between diopside-anorthite compositions [87], 8 self-diffusion of Si
and tracer diffusion Al /Ga [88], 9 tracer diffusion of 24Na and 42K [89], 10Eε is the activation energy of dielectric
relaxation (D.R.) measured in this study, 11 calculated from electrical impedance data [49].
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
32
Fig. 18. Dielectric relaxation time of anorthite glass as a function of pressure. ∆V* is the activation volume calculatedat 800°C according to Eq 22.
The pressure increase results in slowing dielectric relaxation time. This may best be explained by a
continuous diminishment of open paths and mobili ty of alkaline ions, increasing the time for defect
diffusion. The pressure dependence of dielectric relaxation in anorthite is plotted in Fig. 18.The
activation volume of τ for anorthite glass, calculated as follows,
P
EV a
∂∂
=∆ * (22),
is about 10 ± 5 cm³/mol. For albite glass the estimated activation volume is about 0 (-2 ± 1
cm³/mol, Table 5). For comparison, MD simulations [78] of the tracer diffusion in albite provides
for Na an activation volume ~3 cm³/mol, ~ -4.5 cm³/mol for Al, ~ -6 cm³/mol for Si, and ~ -5
cm³/mol for O. From the negative pressure dependence of viscosity the calculated activation
volume of the shear viscosity in albite melt at 1900-2000 K and in pressure range 2.6-5.3 GPa is -
5.4 cm³ mol-1 [2]. It should be noted here that the experimental error of the data in Fig. 18 is rather
large, the data were collected in three different apparatus (atmospheric pressure furnace, piston-
cylinder and belt apparatus), each with different sample geometries.
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
33
Table 6. Prigogine-Defay ratio and pressure dependence of glass transition
temperature.
e f g h i j h k l m g n o oAnorthite SiO2 Albite HPG8 Orthoclase
p q r s t -5 K-1 2.3 [58]
1.9 [47]
3.0 [91]
0.135 [53]
10.3 [90]
1.1 [17]
0.76 [58]
3.3 [91]
1.7 [36] 0.8 [58]
2.4 [92]
u v w10-11 Pa-1 2.5-2 [93]
1.92[95]
5.7 [K86]
6.3 [96]
4-3 [94]
2.12 [95]
3.73 [93]
2.12 [95]
4.5-43 [93]
2.52 [95]x y
p J mol-1 K-1 48 [68]
56.3 [92]
8 [68] 26.7 [68] 8 [60] 21.6 [55]
37 [92]
V z | -6 m3 mol-1 110 [58] 27.4 [K86] 112 [58] 28.2 [36] 1121 [97]
Tg, K 1160 [55]
1109 [48]
1373 [K86]
1460 [68]
1096 [68]
1036 [92]
1129 [H96]
1073 [60]
1221 [55]
1178 [92]
22-1.5 7.6~ 103 – 1.2 135 – 4.2 34-18 200 – 20
p
gg
C
VT
dP
dT
∆∆
=α
,
K GPa-1
185 - 47 470 – 6 150 - 33 70-64 145 – 28
dP
dTg, K GPa-1
[ts]
5
30 (P< 1GPa)
17 -8 -45
1 calculated from [97]; 2 T from [95]; 3 data are
extrapolated from [93], ; [ts] – from electrical conductivity measurements, this study.
Table 6 summarises the literature data on parameters used in calculations of Prigogin-Defay
parameter and the pressure dependence of Tg for studied sil icate glasses. The span of parameter Π
is one order of magnitude and it is much higher than 1 for all sil icate glasses. The extreme high
value of Π= 1.7·104 is calculated for SiO2 by using the thermal expansion coefficient measurements
from [53]. However, if one uses the thermal expansivity data from [90] the value of Π for silica
glass may be even close to 1. In the present work, the upper limit of Π estimated for SiO2 glass by
assuming that ∆α ∼ α from [53]. In reality, the thermal expansion coefficient of sil ica glass is
negative at temperatures up to 100-150° above Tg and, then, above 2000K becomes again positive ~
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
34
10-4 K-1[53]. In [17] α of SiO2 was erroneously averaged in the wide temperature interval and the
suggested ∆α between silica glass and melt ~ 1.1·10-5 K-1 is tool small . In [17] the estimations of Π
for albite and anorthite (from15 to 3 and from 38 to 7, respectively), the authors used rather
arbitrary values of ∆β (the contrast of isothermal compressibilities in glass and in melt has been
taken from 1 to 5·10-11 Pa-1). In the present estimation of Π, the experimental compressibility of
melt was calculated by using the method of [95] and extrapolated data of [93]. The contrast in
compressibility between melt and glass was assumed 0.5βT of the melt. In reality, ∆βT may cover a
much broader span. For example, in metallic glasses ∆βΤ is about 1% of βΤ [19], in SiO2 ∆β is
about 75% of β . For albite glass, the compressibility from the measurements of glass
density at high pressure were also used [94]. A comparison between calculated and measured
dTg/dP in electrical impedance experiments indicates that only SiO2 and anorthite (if only data at
pressures <1 GPa are taken into account) are in a satisfactory agreement. Otherwise, Tg and dTg/dP
estimated from electrical impedance data for alkaline aluminosil icates are very different from the
structural glass transition. But even for anorthite, for which the present estimations of Tg are close
to the structural glass transition, the Π value is much larger than 1, which does not permit
consideration of the glass transition in anorthite as a second order phase transition, i.e. a relaxation
process with one ordering parameter.
The results of Tg estimation from electrical impedance measurements provide different
results in comparison with rheological, calorimetric or dilatometric measurements. There are few
reasons for that. The question which arises, is it right to compare dynamic methods of Tg
determination using variable frequency and methods using down- and up-scan definitions of Tg [1].
Usually, dynamic or ac methods result in higher values of Tg in comparison of time scanning of
heating-cooling cycles. The observed in the present study small difference of Tg for anorthite and
SiO2 may be attributed to the difference in the definition of Tg: as midpoint of specific heat capacity
rise in a heating temperature scan and as a temperature of the maximum in imaginary part of the
electrical impedance. For alkaline bearing glasses the difference in Tg obtained from electrical
measurements and rheological, calorimetric or dilatometric studies, from other hand, is more
fundamental. Essentially, the difference in the relaxation time of mechanical and electric properties
lies in the differing nature of contributions of these two relaxation and may attain a factor of 101-2
near Tg for some glass formers [37]. To electric relaxation there is a contribution from two
processes, charge separation and charge migration. Thus, it may expected that in glass where
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
35
charge separation is significant, will behave differently from glasses where charge transport is
mostly due to a charge carrier migration. In sil icate glasses and melt containing in the structure Na+
the effects of charge separation are very important, sodium is highly unharmonic species,
especially, in the aluminosil icate structure. Sodium ions are responsible for a significant
polarisation (n<<1 in Eq. 17). By replacing Si4+ to Al3+ sodium becomes surrounded by several
non-bridging oxygen ions and aluminium tetrahedra which results in an increase of the dielectric
constant and increase of ionic polarasabil ity with the increase of aluminium substitution [80]. The
albite and haplo-granitic glasses due to their high conductivity and polarasabil ity can be classified
as “ loose” glass formers. The Tg measured in these glasses by electrical impedance method is a
temperature above which Na+ ions become sufficiently mobile, and the kink in the slope log(σ) vs.
1/T,K corresponds to “mobile Na-ion Tgs[81].
Judging from the sign of dTg/dP for studied glasses and qualitative behavior of the fragil ity
index m from Eq 14 or 16, it may be concluded that “ loose” conductive glasses (albite and
haplogranite) becomes even more fragile with the pressure increase, and “tight” glasses (anorthite
and SiO2) are getting more stronger under pressure.
Conclusions
1. The use of electrical impedance measurements to determine a glass transition temperature at
high pressures provides results for anorthite and sil ica glasses consistent with previous studies.
According to the structure of these glasses, the transport or diffusion of structural and
electrically charged defects are correlated. Anorthite may be classified as a “tight” fragile glass
former, silica glass is probably a “tight” strong glass. For albite and haplogranitic glasses the
established Tg is not a structural glass transition because of the “decoupling” between alkaline
mobil ity/charge separation and structural relaxation. These glasses belong to a class of “ loose”
glassy conductors.
2. Pressure dependence of Tg estimated from electrical conductivity measurements reflects
viscosity variations with pressure. For albite, the negative slope dTg/dP corresponds to a
decrease of the shear viscosity of albite melt under pressure. The greatest variation of Tg with
pressure occurs at P< 1 GPa. Further decrease of Tg is hindered by a decrease of percolation
paths for conduction. As a result of this, the contrast between electrical conductivity below and
above Tg becomes indistinguishable at pressures above 1 GPa.
Bagdassarov et al. „Pressure dependence of Tg “ submitted to „Physics and Chemistry of Glasses“
36
3. Effect of pressure on dielectric relaxation times and the activation energy dielectric relaxation
may be explained in terms of the activation volume of dielectric relaxation. Positive activation
volume relates to a positive dTg/dP and strong behavior under pressure (SiO2, anorthite).
Negative or 0 activation volume correlates with the negative dTg/dP (sodium-bearing sil icates).
4. The determined dTg/dP values compared with those calculated from the second Ehrenfest
relationship show a significant discrepancy. Combined with the fact that a significant deviation
of the Prigogin-Defay ratio from 1 is observed, this indicates a difference of the glass transition
in sil icates from a second order phase transition.
Acknowledgements
The experimental works has been supported by Deutsche Forschungsgemeinschaft in the frame of
Special Program „Melts in the Earth: structure, formation and properties“ (SPP 1015). The authors
grateful to H. Höfer and T. Stachel for the help with the microprobing. Multi-anvil experiments
were performed at Bayerisches Geoinstitut under the EU “TMR – Large Scale Facil ities“
programme (Contract No. ERBFMGECT980111 to Prof. D. C. Rubie). A. S. and V. B. thank
Exchange Program between DFG and Russian Academy of Sciences for covering their travel and
stay costs at Univ. Frankfurt.
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