# Viscosity and Glass Transition in Amorphous Oxides

Authors: Michael I. Ojovan
Journal: Advances in Condensed Matter Physics
 (2008)

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## Abstract

An overview is given of amorphous oxide materials viscosity and glass-liquid transition phenomena. The viscosity is a continuous
function of temperature, whereas the glass-liquid transition is accompanied by explicit discontinuities
in the derivative parameters such as the specific heat or thermal expansion coefficient.
A compendium of viscosity models is given including recent data on viscous flow model
based on network defects in which thermodynamic parameters of configurons—elementary
excitations resulting from broken bonds—are found from viscosity-temperature relationships.
Glass-liquid transition phenomena are described including the configuron model of
glass transition which shows a reduction of Hausdorff dimension of bonds at glass-liquid transition.


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## Body

## 1. Introduction

Solids can be either amorphous or crystalline in
structure. In the solid-state, elementary particles (atoms, molecules), which
form the substance, are in fixed positions arranged in a repeating pattern in
crystalline solids or in a disordered pattern in amorphous solids. The
structures of crystalline solids are formed of repeating  regular units, for example, unit cells. Each unit cell of a crystal is defined
in terms of lattice points, for example, the points in space about which the
particles are free to vibrate. The structure of amorphous materials cannot be
described in terms
of repeating unit cells; because of nonperiodicity, the unit cell of an
amorphous material would comprise all atoms. Both solid-state physics and
chemistry focus almost entirely on crystalline form of matter [1–3], whereas the
physics and chemistry of amorphous state in many aspects remain poorly
understood. Although numerous experiments and theoretical works have been
performed, many of the amorphous-state features remain unexplained and others
are controversial. One of such controversial problems is the nature of
glass-liquid transition. The difficulty in treating the
glass transition is caused by almost undetectable changes in the structure despite the qualitative changes
in characteristics and extremely large change in the time scale [4]. The
translation-rotation symmetry of particles is unchanged at the liquid-glass
transition, which retains the topological disorder of fluids. Like a liquid, a
glass has a topologically disordered distribution of atoms and molecules but
elastic properties of an isotropic solid. The symmetry similarity of both
liquid and glassy phases leaves unexplained qualitative differences in their behaviour.
We demonstrate below that there is a qualitative difference in the symmetries
of liquid and glasses when the system of joining bonds is examined rather than
the distribution of material elementary particles.According to the nature of the bonds which hold
particles together, condensed materials can be classified as metallic, ionic,
molecular, or covalent network solids or fluids. For example, the principal
types of binding are caused by collective electrons in metals, electrostatic
forces between positive and negative ions in ionic materials, overlapping
electron distributions in covalent structures, and van der Waals forces in
molecular substances [2]. One of the useful approaches is to consider the bond
system instead of considering the set of atoms or molecules that form the
matter. For each state of matter, we can define the set of bonds, for example,
introduce the bond lattice model which
is the congruent structure of its chemical bonds. The congruent bond
lattice is a regular structure for crystalline materials and disordered for
amorphous materials. A configuron is defined as an elementary
configurational excitation in an amorphous material which involves breaking of
a chemical bond and associated strain-releasing local adjustment of centres of
atomic vibration. The higher the temperature of an amorphous material is, the higher the
configuron concentration is.
Configurons weaken the bond system, so that the higher the content of
configurons is, the
lower the viscosity of an amorphous material is. At very high concentrations, configurons form
percolation clusters. This means that the material loses its rigidity as it becomes penetrated
by a macroscopic (infinite-size) clusters made of broken bonds. The formation
of percolation clusters made of configurons gives an explanation of glass transition
in terms of percolation-type phase transitions [5]. Moreover, although no symmetry changes can be revealed in the atomic distribution,
there is a stepwise variation of Hausdorff dimension of bonds at the glass
transition, namely, there is a reduction of Hausdorff dimension of bonds from
the 3 in the glassy state to the fractal df=2.55±0.05 in the liquid state [5, 6].We give
an overview of viscosity and glass-transition phenomena in amorphous oxides in
this paper. Viscosity-temperature relationships and viscosity models are
considered including recent data on viscous flow model based on network defects
in which thermodynamic parameters of configurons—elementary
excitations resulting from broken bonds in amorphous oxides—are found from viscosity-temperature
relationships. Glass-liquid transition phenomena are described along with the
configuron model of glass transition in which the Hausdorff dimension of
material bond system changes at glass transition.


## 2. Amorphous Oxide Materials

Oxides
are the most abundant inorganic substances on the Earth and at the same time
among the most important materials for practical applications. Oxide materials are of
excellent environmental stability [7, 8]. They are used in many applications
from home cookware to industrial thermal insulations and refractories as well
as nuclear waste immobilisation [9]. Oxide materials exist in a
wide variety of chemical compositions and crystal structures. However, oxide materials
are most frequently found in disordered, for example, amorphous state. There is an enormous diversity of amorphous materials, including
covalently-bonded oxide glasses such as vitreous silica, the structure of which
is modelled by a continuous random network of bonds (network-forming materials);
metallic glasses bonded by isotropic pair potentials, whose structure is
thought of as a dense random packing of spheres; amorphous polymers, whose
structure is assumed to be an arrangement of interpenetrating random-walk-like
coils strongly entangled with each other. Silicate glasses are representative examples
of widely used oxide amorphous materials with everyday importance.Although disordered, the oxide materials in
the most frequently used glassy state exhibit similar mechanical properties to
crystalline materials.
The International Commission on Glass defines
glass as a uniform amorphous solid material, usually produced when the viscous
molten material cools very rapidly to below its glass transition temperature,
without sufficient time for a regular crystal lattice to form [10]. The IUPAC Compendium on Chemical Terminology
defines glass transition as a second-order transition in which a supercooled
melt yields, on cooling, a glassy structure [11]. It states that below the
glass-transition temperature, the physical properties of glasses vary in a
manner similar to those of the crystalline phase. Moreover, it is deemed that the bonding structure of glasses although
disordered has the same symmetry signature (Hausdorff-Besikovitch
dimensionality) as for crystalline materials [6, 12].Glass is one of the most ancient of all
materials known and used by mankind. The natural glass, obsidian, was first
used by man thousands of years ago to form knives, arrow tips, and jewellery.
Manmade glass objects from Mesopotamia have been dated as early as 4500 BC and
from Egypt from 3000 BC. The high chemical resistance of glass allows it to
remain stable in corrosive environments for many thousands and even millions of
years. Several glasses are found in nature such as obsidians (volcanic
glasses), fulgarites (formed by lightning strikes), tektites found on land in
Australasia and associated microtektites from the bottom of the Indian Ocean,
moldavites from central Europe, and Libyan Desert glass from western Egypt. Some
of these glasses have been in the natural environment for about 300 million
years with low alteration rates of less than a millimetre per million years. For example, the natural glass obsidian
is formed when lava erupts from volcanoes and cools rapidly without sufficient
time for crystal growth. The composition of a typical California
obsidian is (wt%) 75SiO2 13.5Al2O  1.6FeO/Fe2O3 1.4CaO 4.3Na2O 4.5K2O 0.7MnO. Obsidian glass edges can be extremely sharp reaching
almost molecular thinness and was known for its ancient use as knives and
projectile tips. Tektites are
other natural glasses, typically up to a few centimetres in size, which have
most probably been formed by the impact of large meteorites on the Earth
surface which melted the Earth surface material resulting on cooling in glass.
The age of tektites found in Czech Republic, moldavites of typical composition
(75–80)SiO2 (9–12)Al2O  (1–3)FeO/Fe2O3 (2-3)CaO 0.3Na2O  3.5K2O,  is assessed to be approximately 15
million years [7].Glasses
are most frequently produced by a melt cooling below its glass-transition
temperature sufficiently fast to avoid formation of crystalline phases.
Glass-forming materials such as dioxides do not require very fast cooling,
whereas quickly crystallising materials such as metals require a very fast
cooling (quenching), for example, the early metallic glasses had to be cooled
extremely rapidly~106 K/s  to avoid crystallisation [13, 14]. Glasses can be formed by several methods
such as(i)
melt quenching [7];(ii)
physical vapour deposition [15];(iii)
solid-state reactions
(thermochemical [16] and mechanochemical [17] methods);(iv)
liquid-state reactions
(sol-gel method [18, 19]);(v)
irradiation of crystalline
solids (radiation amorphisation [20, 21]);(vi)
under action of high pressures
(pressure amorphisation [22, 23]). Glass
formation from melts is a matter of bypassing
crystallisation,
and formation of glass is easier in more complex systems. Oxide glasses
containing a variety of cations are easier to be obtained in a glassy state as
their complexity necessitates longer times for diffusion-controlled
redistribution of diverse constituents before crystallisation can begin. The
vast bulk of glasses used in commerce are oxide glasses. It is assessed that
better than 95% of the commercial tonnage is oxide glasses, of which ~95% is
silica-based glasses [5]. Table 1 gives the composition of important oxide
glasses [7, 24, 25].Table 1
Commercial oxide glass compositions.
Glass familyOxide, mass %(application)SiO2Na2OCaOAl2O3MgOB2O3BaOPbOK2OZnOVitreous
silica(Furnace tubes, Si100melting crucibles)Soda-lime silicate:Window72.014.210.00.62.5trace0.6Container74.015.35.41.03.70.6Bulb and tube73.316.05.21.33.5Tableware74.018.07.50.5Sodium
borosilicate:Chemical glassware81.04.52.012.0Waste immobilisation43–536–240–143–190–5.38–17misc.misc.misc.misc.Lead-alkali
silicate:Lead “crystal”59.02.00.425.012.01.5Television funnel54.06.03.02.02.023.08.0Aluminosilicate:Halogen lamp57.00.0110.016.07.04.06.0traceFibreglass “E”52.917.414.54.49.21.0Optical (crown)68.98.810.12.88.41.0


## 3. Melting of Amorphous Solids

Crystalline
materials melt at well-defined melting temperaturesTm  whereas
amorphous materials transform from glassy, for example, solid form to liquid-state
at glass transition temperatures Tg which in this sense play the
role of melting temperatures for non-crystalline solids. Although fundamentally
important, the nature of the glass transition is not well understood [25–30]. A glass is
most commonly formed by cooling a viscous liquid fast enough to avoid crystallisation.
Compared with  Tm, the actual values of  Tg  depend on
thermal history, for example, cooling rate which makes the understanding of
glass transition phenomena intriguing. The liquid-glass transition is
accompanied by significant changes in physical properties, for example, glasses
are brittle, thus changes should occur at the molecular level although the
material is topologically disordered both in liquid and glassy states.
Rearrangements that occur in an amorphous material at the glass transition
temperature lead to characteristic discontinuities
of derivative thermodynamic parameters such
as the coefficient of thermal expansion [31] or the specific heat (Figure 1).Specific heat of amorphouso-terphenil. (a) Experimental and (b) calculated [12].
(a)(b)These discontinuities allow detecting
theTg  [31] or, accounting for cooling rate dependences, the glass transition
interval where a supercooled liquid transforms to a glass (Figure 2).Figure 2
Determination of glass transition temperatureTg based on differential temperature analysis (DTA), below the Tg the
material is glassy whereas above the Te the material is liquid (after [32]).Vitrification manifests itself as a second-order
phase transition, however its description in terms of the Landau theory is difficult
as there is no clarity about the order parameter describing this transition [33, 34]. Although similar to a second-order phase
transformation, the glass-liquid transition is a kinetically controlled
phenomenon which exhibits a range of Tg depending on the cooling
rate with maximal Tg at highest rates of cooling [31]. In practice,
the liquid-glass transition has features both in common with second-order
thermodynamic phase transitions and of kinetic origin. Glass-forming material
commonly exhibits two types of relaxation process: fast β relaxation (Johari-Goldstein) and slow α relaxation [29, 30]. The attention of the majority of researchers in the last decades has
been focused on relaxation aspects of the liquid-glass transition rather than
the structure [29]. Emphasis in these works is placed on glass nonergodicity
and it is considered that glass is a material characterised by large Deborah
numbers [35] for which the relaxation time is much longer than the observation
time taken typically as 102 seconds. The systems are commonly
assumed to be ergodic at temperatures T>Tg, whereas the systems are
completely frozen with respect to primary relaxation at T<Tg [29, 36]. Figure 3(a) shows
three regions of amorphous materials behaviour based on the temperature
variations of extensive thermodynamic parameters, for example, enthalpy (H) and
specific volume (V) [29]: (a) ergodic, (b) transition, and (c) nonergodic. Goldstein
found that diffusion in liquids occurs by different mechanisms at high and low
temperatures. Jump potential energy barriers are large at low temperatures,
while at high temperatures molecules move almost freely since thermal energies
overcome the barrier heights [37]. The correlation between the energy landscape
and the fragility of liquids was
emphasised by Angell [38]. Recent works [5, 6, 12] have revealed that in
addition to changes in relaxation behaviour, significant changes occur within
the system of bonds at glass-liquid transition (Figure 3(b)).(a) Regions of relaxation behaviour of amorphous materials (after [29]); (b)
density and geometry of bonds on increase of temperature (after [6]).
(a)(b)As
on cooling, the viscosities of glass-forming liquids continuously increase and
achieve very high values, the liquid-glass transition is often regarded as a
transition for practical purposes rather than a thermodynamic phase transition
[2, 39]. By general agreement, it is considered that a liquid on being cooled
becomes practically a glass when the viscosity equals 1012 Pa⋅s (1013 poise) or where the relaxation time is 102 seconds [2, 39]. There is
no phase transformation at this practical purpose (relaxation) glass-transition
temperature [2] which is found from the viscosity-temperature relationship: (1)η(Tg,relax)=1012(Pa⋅s). Despite
the fact that a glass like a liquid has a topologically disordered
structure, at the same time, it has elastic properties of an isotropic solid. Changes should thus occur at the molecular level although the material
is topologically disordered both in liquid and glassy states. The
difficulty is to specify how the structure of a glassy material differs from
that one of a liquid. The translation-rotation
symmetry in the distribution of atoms and molecules is broken at
crystallisation but remains unchanged at the liquid-glass transition, which
retains the topological disorder of fluids.What kind of symmetry is changed at glass-liquid
transition? Amorphous
materials have no elementary cell characterised by a certain symmetry, which
can reproduce the distribution of atoms by its infinite repetition. Instead the
symmetry of a topologically disordered system is characterised by the
Hausdorff-Besikovitch dimensionality of the system bonds. Formally, the Hausdorff dimension of a subsetA of a metric
space X is the infimum (e.g.,
the greatest element, not necessarily in the subset, that is less than or equal
to all other elements of the subset) of d≥0 such
that the d-dimensional Hausdorff measure of A is 0 (which need not be integer)
[40]. In practice, the Hausdorff dimension is defined using the standard
procedure of covering the subset considered (congruent bond structure in our
case) with a number N(λ) of
spheres of radius λ  (or cubes with size λ). In condensed matter, the smallest size
possible for λ  is the bond size. The smaller the λ  is, the
larger the N(λ) is,
and the fractal dimension is found as the limiting case from (2)df=−limλ→0logN(λ)logλ. Roughly,
if N(λ) grows
proportionally to 1/λd  as λ  tends to zero, then we say that the subset has
the Hausdorff dimension d [41]. Using such a procedure, we could find that
3-dimensional bond structures, which have joining bonds intact, are
characterised by the Hausdorff dimension d = 3. This conforms to the known result
that the Euclidian space Rn has the Hausdorff dimension, d = n.
Moreover, whether the distribution of bonds is ordered or disordered is unimportant,
in both cases the 3-dimensional bond structures have the same Hausdorff
dimension, d = 3. Thus, in both cases, for example, in both glasses and crystals
we have the same Hausdorff dimension of the bond system.Two
types of topological disorder characterised by different symmetries can be
revealed in an amorphous material based on the analysis of broken bond
concentrations [5, 6, 12]: (i) 3-dimensional, 3D (Euclidean, d = 3), which occurs
at low temperatures when the configurons are uniformly distributed within the
bond structure with no percolation clusters of configurons formed and the
geometrical structures of bonds can be characterised as a 3D and (ii) df=2.55±0.05-dimensional (fractal), which occurs at high temperatures at least
near the glass transition temperature when percolation clusters made of broken
bonds are formed and the geometries of the dynamic structures formed can be characterised
as fractal objects with preferential pathways for configurons. Hence,
the bonding structure of glasses has the same
Hausdorff-Besikovitch dimensionality (symmetry signature) as for crystalline
materials (Figure 3(b)) whereas the liquid near the glass transition is a
dynamic uniform fractal. We will first consider herein the viscosity of
amorphous materials and then analyse models of glass-liquid transition. Although we do not directly link the melting of amorphous materials (e.g.,
glass-liquid transition) with their viscosity (see (1)), we will use the
viscosity-temperature relationships to identify the thermodynamic parameters of
configurons.


## 4. Viscosity-Temperature Relationships

The
viscosities of fluids are among their most important properties. Viscosity
quantifies the resistance of fluids to flow and indicates their ability to
dissipate momentum. The momentum balance of the Newtonian fluids is described
at the macroscopic level by the Navier-Stokes equations. At the microscopic
level, viscosity arises because of a transfer of momentum between fluid layers
moving at different velocities as explained in the Maxwell kinetic theory. In
oxide melts and glasses, viscosities determine melting conditions, working and
annealing temperatures, rate of refining, maximum use temperature, and crystallisation
rate. In geology, the behaviour of magma and hence volcanic eruptions and lava
flow rates depend directly on the viscosities of molten silicates [42, 43].
Table 2 gives viscosities of several amorphous materials.Table 2
Viscosity of some amorphous materials.
MaterialViscosity,
Pa⋅sWater at 25°C0.894 10−3Mercury at 25°C1.526 10−3Olive oil at 25°C8.1 10−2Glycerol at 25°C0.934Glass
batch at melting point10Pitch at 25°C2.3 108Glass at strain point3 1013It
is commonly assumed that shear viscosity is a thermally activated process.
Since the pioneering work of Frenkel [44] fluid viscosity, η(T), has been expressed in terms of an
activation energy Q by (3)η(T)=Aexp(QRT), where
T is temperature in K, R is the molar gas constant, and A is a constant. For amorphous materials,
two different regimes of flow have been identified with melts at high
temperature having lower
activation energy for flow than materials at lower temperatures. Within the low
temperature or high temperature regimes, an Arrhenius dependence of viscosity
is observed and an appropriate activation energy, QH or QL,
respectively, can be defined. Asymptotically, both at low and high temperatures
the activation energy of viscosity is independent of temperature. This pattern
has been observed with a range of melts including silicates, fused salts,
oxides, and organic liquids [42, 43]. Between the high temperature and the low
temperature regimes, the activation energy for flow changes and the viscosity
cannot be described using the Arrhenius-type behaviour, for example, the
activation energy of viscosity varies with temperature.Viscosity directly
governs the relaxation processes in amorphous materials. The Maxwell relaxation
time gives the characteristic relaxation time to attain stabilised parameters
of a material:(4)τM=ηG, where G is the shear modulus. The higher the
viscosity is, the longer the relaxation times are. Near the glass-transition temperature, the elasticity modulus of a glass
G~1010 Pa [45], hence at η = 1012 Pa⋅s, where the
practical purpose (relaxation) glass transition occurs (see (1)), the Maxwell relaxation time τM~102 seconds.
Accounting that for
fused silica the activation energy of viscosity at low temperatures QH = 759 kJ/mol and the shear modulus of fused silica is 31 GPa at 25°C  [46, 47],
one can see that the relaxation time at STP becomes as long as τM~1098 years which
incommensurably exceeds the lifetime of Universe (approx. 1.5 × 1010 years). This shows again that the glass should be considered a true solid
material [48].The viscosity of amorphous materials depends on
chemical composition, for example, in silicate systems viscosity attains the
highest values for vitreous silica (Figure4).Figure 4
Viscosity of amorphous silicates and important technological points in glass
manufacture industry (after [49]).The more or less randomness, the openness, and
the varying degree of connectivity allow the glass structure to accommodate
large variations in composition, for example, glass acts like a solution.
Moreover, it was found that melts and glasses produced from them can be often
considered as solutions consisting of salt-like products of interactions
between the oxide components [50]. These associates are similar to the
crystalline compounds which exist in the phase diagram of the initial oxide
system. Calculations in this model are based on solving the set of equations
for the law of mass action for the reactions possible in the system of oxides
and the equations of mass balance of the components. This approach describes
well such properties as viscosity, thermal expansion, isothermal
compressibility, and optical parameters [50].For oxide glasses, a small change in glass
composition typically causes a smooth change in glass properties. The unit
addition or substitution of a component can be deemed as a contribution
characteristic of that component to the overall property. This notion gives
rise to the additive relationships with many properties such as densities,
refractive indexes obeying additive relationships [7]. An additive property P
obeys a linear relation of the type (5)P=∑i=1npiCi,where∑i=1nCi=100%, where pi are additivity factors for a given component i=1,2,3,…,n, and Ci are the mass% or the mol% of that component in the glass. In oxide glasses, the
density follows additivity primarily because the volume of an oxide glass is
mostly determined by the volume occupied by the oxygen anions, the volume of
cations being much smaller [7]. Additivity relations work over a narrow range
of compositions and additivity coefficients of a given oxide may change from system
to system. Nonlinearities appear when various constituents interact with each
other. Glass properties can be calculated through statistical analysis of glass
databases such as SciGlass [7, 51]. Linear regression can be applied using
common polynomial functions up to the 2nd or 3rd degrees. For viscosities of
amorphous oxide materials (melts and glasses), the statistical analysis of
viscosity is based on finding temperatures (isokoms) of constant viscosity log[η(Ti)]=consti, typically
when viscosity is 1.5, 6.6, and 12 (point of practical purpose glass
transition) [51–53]. A detailed
overview on statistical analysis of viscosities and individual oxide
coefficients Ci in isokom temperatures of oxide materials is given
in [53]. Addition of oxides to certain base compositions changes the viscosity
and the impact from different oxides is different [5, 51–54]. Figure 5 shows
the effect of component addition to the base composition in mol% 73SiO2 2B2O3 2Al2O3 12Na2O  2K2O
2MgO  7CaO on the temperature where the viscosity log(η(T))=1.5 (isokom) [53].Effect of component addition on isokom log[η(T)/Pa⋅s] = 1.5 [53]. Courtesy
Alexander Fluegel.
(a)(b)


## 5. Fragility Concept

As
noted above, the term glass transition temperature is often used to refer to
the temperature at which the viscosity attains a value of 1012 Pa⋅s.
This definition of  Tg was used by Angell to plot the logarithms of
viscosity as a function of (Tg/T) [55, 56]. In such a plot, strong
melts, that is,  melts that exhibit
only small changes in the activation energy for flow with temperature, such as
silica or germania, have a nearly linear dependence on the inverse of the
reduced temperature whereas fragile melts deviate strongly from a linear
dependence as the activation energies of fragile liquids significantly change
with temperature. However, this change is characteristic only for intermediate
temperatures and the viscosity has asymptotically Arrhenius-type behaviour both
at high and low temperatures. Within the low temperature, the activation energy
of viscosity is high QH whereas at high temperatures the activation
energy is low QL. As asymptotically, both at low and high
temperatures the activation energies of viscosity are independent of
temperature changes that occur in the activation energy can be unambiguously
characterised by the Doremus fragility ratio [42, 43]: (6)RD=QHQL. The Doremus
fragility ratio ranges from 1.45 for silica to 4.52 for anorthite melts (Table 3).Table 3
Asymptotic Arrhenian activation energies for viscosity and the corresponding
Doremus fragility ratios [47].
Amorphous materialQL,
kJ/molQH, kJ/molRDSilica
(SiO2)5227591.45Germania (GeO2)2724011.4766.7SiO2 33.3PbO2744711.7280SiO2 20Na2O2073621.7565SiO2 35PbO2574881.959.9SiO2 40.1PbO2584941.9175SiO2 25Na2O2034362.1575.9SiO2 24.1PbO2345062.16SLS:
70SiO2 21CaO 9Na2O2936342.16Salol
(HOC6H4COOC6H5)1182632.2370SiO2 30Na2O2054632.2665SiO2 35Na2O1864862.61α-phenyl-o-cresol (2- Hydroxydiphenylmethane)1032752.6752SiO2 30Li2O  18B2O31946143.16B2O31133713.28Diopside
(CaMgSi2O6)24010844.51Anorthite
(CaAl2Si2O8)25111354.52The
higher the value ofRD is, the more fragile the melt is. The
fragility of amorphous materials numerically characterised by the Doremus
fragility ratio classifies amorphous materials as strong if they have RD<2, and fragile materials if they have RD≥2.
The implication of strong-fragile classification was that strong fluids are
strongly and fragile are weakly bonded [56]. As pointed out by Doremus [42, 43],
these widely and convenient terms are misleading, for example, binary silicate
glasses are strong although have many nonbridging oxygens. Some network melts
such as anorthite and diopside have very high activation energies being quite
strongly bounded but are very fragile. Nevertheless, the fragility concept
enables classification of melts based on their viscosity behaviour. Those melts
which significantly change the activation energy of viscosity are fragile and
those which have small changes of activation energy of viscosity are
strong.


## 6. Viscosity Models

Many
different equations to model the viscosity of liquids have been proposed. The
first one is the Frenkel-Andrade model which assumes that viscosity is a
thermally activated process described by a simple exponential equation (3) with
constant activation
energy of viscosity [44]. As this simple model fails to describe the behaviour
of viscosity at intermediate temperatures between strain and melting points
(see Figure 4), many other models were developed some of which become popular
and are being extensively used. Although it is well known [57, 58] that the
best description of viscosity is given by the two-exponential equation derived
by Douglas [59], the most popular viscosity equation is that of Vogel, Tamman
and Fulcher (VTF). It gives an excellent description of viscosity behaviour,
namely, at intermediate temperatures which are very important for industry. It
is also most useful in describing the behaviour of amorphous materials in the
transition range (range B on Figure 3(a)), where solidification of amorphous
materials occurs. Adam-Gibbs (AG) and Avramov-Milchev (AM) models are also
often used to describe the viscosity in the intermediate range of temperatures.
Out of the intermediate range, none of these models correctly describe the
behaviour of viscosity and in the limits of low and high temperatures the best
description of viscosity provides the Frenkel-Andrade model with high and low
activation energies. Moreover, there is a tendency to a nonactivated regime of
viscosity of melts at very high-temperatures [47]. Figure 6 summarises the
temperature behaviour of viscosity within various temperature intervals and
indicates the character of activation energy of viscosity with best equations
to be used including the equation of viscosity valid at all temperatures (see below  (12)).Figure 6
Temperature behaviour of viscosity of amorphous materials as described
by the universal equation (12). The activation energy of viscous flow is
constant and high at low temperatures, it is variable at intermediate
temperatures, and it is constant and low at high temperatures. At extreme high
temperatures, the flow becomes nonactivated.


### 6.1. Vtf Model

The
VTF equation of viscosity is an empirical expression which describes viscosity
data at intermediate temperatures (see Figure6) over many orders of magnitude
with a high accuracy: (7)ln[η(T)]=AVTF+BVTFR(T−TV), where AVTF, BVTF, and TV  (Vogel temperature) are constants determined by fitting (7) to experimental
data. Although perfectly working at intermediate temperatures at high and low
temperatures, (7) does not describe the experimental temperature dependence of
viscosity.The
VTF equation can be derived from the free volume model which relates the
viscosity of the melt to free (or excess) volume per moleculeVf.
The excess volume is considered to be the specific volume of the liquid minus
the volume of its molecules. This molecular volume is usually derived from a
hard sphere model of the atoms in the molecules. Molecular transport is
considered to occur when voids having a volume greater than a critical value
form by redistribution of the free volume [60]. The flow unit or molecule is imagined
to be in a structural cage at a potential minimum. As the temperature increases,
there is an increasing amount of free volume that can be redistributed among
the cages, leading to increased transport and this leads to an exponential
relationship between viscosity and free volume [60]: (8)η=η0exp(BV0Vf), where V0 is the volume of a molecule, η0 and  B are constants. In terms of the
specific volume V per molecule, it
can be shown that Vf=V−V0=V0(T−T0)/T0,
for some constant and low temperatures T0. Clearly, (8) is the
same as (7) when AVTF=lnη0,  BVTF=BT0  and TV=T0.
The generic problem with the free volume theory is that the specific volume
of a liquid as a function of temperature shows a discontinuity in slope at the
glass transition temperature (see Figure 3(b)), whereas the viscosities of
liquids show no discontinuities at the glass transition temperature [43].


### 6.2. Adam and Gibbs Model

The
Adam and Gibbs (AG) equation is obtained assuming that above the glass
transition temperature molecules in a liquid can explore many different
configurational states over time, and that as the temperature is raised higher
energy configurational states can be explored. In contrast, below the glass
transition temperature it is assumed that the molecules in the glass are
trapped in a single configurational state. The resulting AG equation for
viscosity is similar to the VTF equation [61]: (9)ln[η(T)]=AAG+BAGTSconf(T),   where AAG  and BAG are adjustable constants
and Sconf(T) is the configurational entropy.
Assuming that Sconf(T)=ΔCp(T−TV)/T,
where ΔCp is the relaxational part
of the specific heat, one can see that  (9) transforms into VTF (7), where AAG=AVTF and BAG=ΔCpBVTF/R.
The configurational entropy model of Adam and Gibbs fits a large number of
viscosity data but like the free volume theory, it does not provide an accurate
fit over the entire temperature range. At high and low viscosities, (9) does
not describe the experimental temperature dependence of viscosity and
increasingly large deviations from the experimental values are produced. In
addition, the configurational entropy model gives discontinuities in the first
differential of the entropy at the glass transition, despite the fact that that
there are no discontinuities in experimentally measured viscosities, for
example, the problem with the entropy theory is the same as for the free volume
theory [43].


### 6.3. Avramov and Milchev Model

The
Avramov and Milchev (AM) viscosity model describes
the viscosity behaviour within the temperature range, where the activation
energy of viscosity changes with temperature (see Figure6). The AM model assumes that due to existing disorder, activation
energy barriers with different heights occur and that the distribution function
for heights of these barriers depends on the entropy. Thus, viscosity is
assumed to be a function of the total entropy of the system which leads to the
stretched exponential temperature dependence of equilibrium viscosity  [45, 62]: (10)ln[η(T)]=AAM+2.3(13.5−AAM)(TgT)α, where
in this case Tg  is defined
by ln⌊η(Tg)/(dPa⋅s)⌋=13.5, AAM  is a constant, and α is the Avramov fragility parameter. The higher α is, the less strong is a fluid so that strong
liquids have a value of α close to
unity. One should note, however, that (10) fails to describe the experimental
temperature dependence of viscosity in the limits of high and low temperatures.


### 6.4. Two-Exponential Equation

The above equations can only be used within limited temperature
ranges that essentially correspond to the range of temperatures, where the activation energy for flow changes with temperature. None
of them correctly
describes the
asymptotic low and high temperature Arrhenian viscosity behaviour [42, 43]. In
addition, the nonphysical character of the fitting parameters does not give a
clear understanding of changes that occur with temperature or composition.
Therefore, these equations may be useful for fitting experimental measurements
over limited temperature ranges, but they cannot explain the temperature
dependencies of viscosity. It is well known [43, 58] that, mathematically, the
viscosity of amorphous materials can most exactly be described by the
two-exponential equation (11)η(T)=ATexp(BRT)[1+Cexp(DRT)], where A, B, C,  and D are all constants. This equation has been derived by Douglas for silicate glasses by assuming that the oxygen
atoms between two silicon atoms could occupy two different positions, separated
by an energy barrier [59] with flow being limited by the breaking of Si–O–Si
bonds. In addition to the fact that (11) provides a very good fit to the
experimental data across the entire temperature range, it correctly gives
Arrhenian-type asymptotes at high and low temperatures with QH=B+D and QL=B.
For the low viscosity range (log(η/dPa⋅s)<3), Volf gives QL = 80–300 kJ/mol and for the high viscosity range log(η/dPa⋅s)>3 and QH = 400–800 kJ/mol [58]. Moreover, within narrow temperature intervals, (11) can
be approximated to many types of curves, such as those given by (7) and (10).
However, in contrast to them (11) gives a correct asymptotic Arrhenius-type
dependence of viscosity with temperature at low and high temperatures when the
activation energy of viscosity becomes constant. Equation (11) follows
immediately from the Doremus conception of defect-mediated viscous flow [5, 6, 12, 47, 63, 64].


## 6.1. Vtf Model

The
VTF equation of viscosity is an empirical expression which describes viscosity
data at intermediate temperatures (see Figure6) over many orders of magnitude
with a high accuracy: (7)ln[η(T)]=AVTF+BVTFR(T−TV), where AVTF, BVTF, and TV  (Vogel temperature) are constants determined by fitting (7) to experimental
data. Although perfectly working at intermediate temperatures at high and low
temperatures, (7) does not describe the experimental temperature dependence of
viscosity.The
VTF equation can be derived from the free volume model which relates the
viscosity of the melt to free (or excess) volume per moleculeVf.
The excess volume is considered to be the specific volume of the liquid minus
the volume of its molecules. This molecular volume is usually derived from a
hard sphere model of the atoms in the molecules. Molecular transport is
considered to occur when voids having a volume greater than a critical value
form by redistribution of the free volume [60]. The flow unit or molecule is imagined
to be in a structural cage at a potential minimum. As the temperature increases,
there is an increasing amount of free volume that can be redistributed among
the cages, leading to increased transport and this leads to an exponential
relationship between viscosity and free volume [60]: (8)η=η0exp(BV0Vf), where V0 is the volume of a molecule, η0 and  B are constants. In terms of the
specific volume V per molecule, it
can be shown that Vf=V−V0=V0(T−T0)/T0,
for some constant and low temperatures T0. Clearly, (8) is the
same as (7) when AVTF=lnη0,  BVTF=BT0  and TV=T0.
The generic problem with the free volume theory is that the specific volume
of a liquid as a function of temperature shows a discontinuity in slope at the
glass transition temperature (see Figure 3(b)), whereas the viscosities of
liquids show no discontinuities at the glass transition temperature [43].


## 6.2. Adam and Gibbs Model

The
Adam and Gibbs (AG) equation is obtained assuming that above the glass
transition temperature molecules in a liquid can explore many different
configurational states over time, and that as the temperature is raised higher
energy configurational states can be explored. In contrast, below the glass
transition temperature it is assumed that the molecules in the glass are
trapped in a single configurational state. The resulting AG equation for
viscosity is similar to the VTF equation [61]: (9)ln[η(T)]=AAG+BAGTSconf(T),   where AAG  and BAG are adjustable constants
and Sconf(T) is the configurational entropy.
Assuming that Sconf(T)=ΔCp(T−TV)/T,
where ΔCp is the relaxational part
of the specific heat, one can see that  (9) transforms into VTF (7), where AAG=AVTF and BAG=ΔCpBVTF/R.
The configurational entropy model of Adam and Gibbs fits a large number of
viscosity data but like the free volume theory, it does not provide an accurate
fit over the entire temperature range. At high and low viscosities, (9) does
not describe the experimental temperature dependence of viscosity and
increasingly large deviations from the experimental values are produced. In
addition, the configurational entropy model gives discontinuities in the first
differential of the entropy at the glass transition, despite the fact that that
there are no discontinuities in experimentally measured viscosities, for
example, the problem with the entropy theory is the same as for the free volume
theory [43].


## 6.3. Avramov and Milchev Model

The
Avramov and Milchev (AM) viscosity model describes
the viscosity behaviour within the temperature range, where the activation
energy of viscosity changes with temperature (see Figure6). The AM model assumes that due to existing disorder, activation
energy barriers with different heights occur and that the distribution function
for heights of these barriers depends on the entropy. Thus, viscosity is
assumed to be a function of the total entropy of the system which leads to the
stretched exponential temperature dependence of equilibrium viscosity  [45, 62]: (10)ln[η(T)]=AAM+2.3(13.5−AAM)(TgT)α, where
in this case Tg  is defined
by ln⌊η(Tg)/(dPa⋅s)⌋=13.5, AAM  is a constant, and α is the Avramov fragility parameter. The higher α is, the less strong is a fluid so that strong
liquids have a value of α close to
unity. One should note, however, that (10) fails to describe the experimental
temperature dependence of viscosity in the limits of high and low temperatures.


## 6.4. Two-Exponential Equation

The above equations can only be used within limited temperature
ranges that essentially correspond to the range of temperatures, where the activation energy for flow changes with temperature. None
of them correctly
describes the
asymptotic low and high temperature Arrhenian viscosity behaviour [42, 43]. In
addition, the nonphysical character of the fitting parameters does not give a
clear understanding of changes that occur with temperature or composition.
Therefore, these equations may be useful for fitting experimental measurements
over limited temperature ranges, but they cannot explain the temperature
dependencies of viscosity. It is well known [43, 58] that, mathematically, the
viscosity of amorphous materials can most exactly be described by the
two-exponential equation (11)η(T)=ATexp(BRT)[1+Cexp(DRT)], where A, B, C,  and D are all constants. This equation has been derived by Douglas for silicate glasses by assuming that the oxygen
atoms between two silicon atoms could occupy two different positions, separated
by an energy barrier [59] with flow being limited by the breaking of Si–O–Si
bonds. In addition to the fact that (11) provides a very good fit to the
experimental data across the entire temperature range, it correctly gives
Arrhenian-type asymptotes at high and low temperatures with QH=B+D and QL=B.
For the low viscosity range (log(η/dPa⋅s)<3), Volf gives QL = 80–300 kJ/mol and for the high viscosity range log(η/dPa⋅s)>3 and QH = 400–800 kJ/mol [58]. Moreover, within narrow temperature intervals, (11) can
be approximated to many types of curves, such as those given by (7) and (10).
However, in contrast to them (11) gives a correct asymptotic Arrhenius-type
dependence of viscosity with temperature at low and high temperatures when the
activation energy of viscosity becomes constant. Equation (11) follows
immediately from the Doremus conception of defect-mediated viscous flow [5, 6, 12, 47, 63, 64].


## 7. Defect Model of Viscous Flow

Doremus analysed data on
diffusion and viscosity of silicates and suggested that diffusion of silicon and oxygen in these
materials takes place by transport of SiO molecules formed on dissociation of
SiO2. Moreover, these molecules are stable at high temperatures and
typically results from the vapourisation of SiO2 [42, 43]. He concluded
that the extra oxygen atom resulting from dissociation of SiO2  leads to
five-coordination of oxygen atoms around silicon. The
three-dimensional (3D) disordered network of silicates is formed by [SiO4]  tetrahedra interconnected via bridging oxygens   ≡Si•O•Si≡, where • designates a bond between Si and
O, and—designates a bridging oxygen atom with two
bonds •O•. The breaking out of an SiO molecule from the SiO2 network leaves behind three oxygen ions and one silicon ion with unpaired
electrons. One of these oxygen ions can bond to the silicon ion. The two other
dangling bonds result in two silicon ions that are five-coordinated to oxygen
ions. Moreover, one of the five oxygen ions around the central silicon
ion has an unpaired electron, and it is not bonded strongly to the silicon ion
[42, 43]. Doremus suggested that this electron hole (unpaired electron) should
move between the other oxygen ions similar to the resonance behaviour in aliphatic
organic molecules. There is an experimental evidence for five-coordination of
silicon and oxygen at higher pressures in alkali oxide SiO2 melts
from NMR, Raman, and infrared spectroscopy, and evidence for five-coordinated
silicon in a K2Si4O9 glass at atmosphere
pressure [65]. Doremus concluded that in silicates, the defects involved in
flow are SiO molecules resulting from broken silicon-oxygen bonds and therefore
the SiO molecules and five-coordinated silicon atoms involved in viscous flow
derive from broken bonds. Although he failed to reproduce the two-exponential
equation of viscosity [43], it was later shown [63, 64] that (11) is a direct
consequence of Doremus ideas. Indeed, assuming that viscous flow in amorphous
materials is mediated by broken bonds, which can be considered as
quasiparticles termed configurons [64], one can find the equilibrium
concentrations of configurons Cd=C0f(T), where f(T)=exp(−Gd/RT)/[1+exp(−Gd/RT)]  Gd=Hd−TSd is the formation Gibbs-free energy, Hd is the enthalpy, Sd is the entropy, and C0 is the total concentration
of elementary bond network blocks or the concentration of unbroken bonds at
absolute zero. The viscosity of an amorphous material can be related to the
diffusion coefficient, D, of the
configurons which mediate the viscous flow, via the Stokes-Einstein equation η(T)=kT/6πrD, where k is the Boltzmann constant and r is the radius of configuron. The probability of a configuron having the energy
required for a jump is given by the Gibbs distribution w=exp(−Gm/RT)/[1+exp(−Gm/RT)], where Gm=Hm−TSm is the Gibbs-free energy of motion of a
jumping configuron, Hm and Sm are the enthalpy and
entropy of configuron motion. Thus, the viscosity of amorphous materials is
directly related to the thermodynamic parameters of configurons by [5, 6, 12, 47, 63, 64] (12)η(T)=A1T[1+A2exp(BRT)][1+Cexp(DRT)] with (13)A1=k6πrD0,A2=exp(−SmR),B=Hm,C=exp(−SdR),D=Hd, where D0=fgλ2zp0ν0, f is the correlation factor, g is a geometrical factor (approx. 1/6), λ is
the average jump length, ν0 is the configuron vibrational
frequency or the frequency with which the configuron attempts to surmount the
energy barrier to jump into a neighboring site, z is the number of nearest neighbours,
and p0 is a configuration
factor.Experiments show that, in practice, four
fitting parameters usually suffice [47], indicating that viscosity is well
described by (11), which follows from (12) if A2exp(B/RT)≫1 (usually the case) and letting A=A1A2. Equation (12) can be fitted
to practically all available experimental data on viscosities of amorphous
materials. Moreover, (12) can be readily approximated within a narrow
temperature interval by known empirical and theoretical models such as VTF, AG,
or the Kohlrausch-type stretched-exponential law. In contrast to such
approximations, (12) can be used in wider temperature ranges and gives correct
Arrhenius-type asymptotes of viscosity at high and low temperatures. Equation
(12) also shows that at extremely high temperatures when T→∞, the
viscosity of melts changes to a nonactivated, for example, non-Arrhenius behaviour, which is characteristic of
systems of almost free particles.The five coefficientsA1, A2, B, C,  and D in (12) can be treated as fitting
parameters derived from the experimentally known viscosity data. Having
obtained these fitting parameters, one can evaluate the thermodynamic data of
configurons (e.g., network breaking
defects) [47]. Hence, from known viscosity-temperature relationships of
amorphous materials one can characterise the thermodynamic parameters of
configurons. As the number of parameters to be found via fitting procedure is
high (5 parameters when using (12) or 4 parameters when using (11)) and both
equations are nonlinear, dedicated genetic algorithm was used to achieve the
best fit between either (11) or (12) and experimental viscosity data [47]. An
example of such evaluation is demonstrated in Figure 7, which shows the best
fit for viscosity-temperature data of amorphous anorthite and diopside obtained
using (12).(a) Viscosity-temperature data for
anorthite and (b) for diopside, where calculated curves were obtained using
(12).
(a)(b)Calculations show that the description of
experimental data using (11) is excellent with very low and uniformly scattered
deviations. Although (11) provides a very good description of the viscosity
data of most melts, it was found that for soda-lime silica system (mass%):
70SiO2  21CaO  9Na2O  and B2O3 at very
high temperatures it gives slightly, but systematically lower results compared
experimental data. Thus, viscosities of these two materials at very high
temperatures are better described using the complete equation (12) rather than
(11) [47].Using relationships (13) from the numerical
data of evaluated parameters A1,
A2, B, C,  and D which provide the best fit of theoretical viscosity-temperature
relationship (12) to experimental data, we can find enthalpies of formation and
motion and entropies of formation and motion of configurons (bond system) of
amorphous materials [47]. Evaluated thermodynamic parameters (enthalpies and
entropies of formation and motion) of configurons in a number of amorphous
materials are given in Table 4.Table 4
Thermodynamic parameters of configurons.
Amorphous materialHd/ kJ mol−1Hm, kJ mol−1Sd/RSm/RSd/SmSilica (SiO2)23752217.5411.371.54SLS
(mass %): 70SiO233129344.0324.401.821CaO  9Na2O80SiO2 20Na2O15520717.987.792.3166.7SiO2 33.3PbO19727425.407.33.4865SiO2 35PbO23125730.328.533.5559.9SiO2 40.1PbO23625831.126.554.6B2O325811344.29.214.865SiO2 35Na2O30018640.717.595.3670SiO2 30Na2O25820534.845.225.8775.9SiO2 24.1PbO26223436.255.446.66Germania (GeO2)12927217.772.497.1475SiO2  25Na2O23320330.624.227.26Anorthite
(CaAl2Si2O8)88425179.550.37421352SiO2 30Li2O 18B2O342019452.060.227229Salol14511868.130.114598(HOC6H4COOC6H5)α–phenyl–o–cresol17210383.840.134626Diopside
(CaMgSi2O6)83424088.710.0442016Data from Table4 show that for most materials
examined the entropy of configuron formation is significantly higher than the
entropy of configuron motion Sd≫Sm. Taking into account the values of Hm this means that Gm/RT≫1 and thus it is legitimate
to simplify (12) to the simpler equation (11), that is, four fitting parameters are usually sufficient to
correctly describe the viscosity-temperature behaviour of a melt. Notable
exception is the SLS glass considered (mass%: 70SiO2 21CaO 9Na2O)
which, at high temperatures, exhibits deviations from (11) and requires (12) [47].


## 8. Glass-Liquid Transition

Amorphous materials can be either liquid at
high temperatures or solid, for example, glassy or vitreous solids at low
temperatures. The transition from the glassy to the liquid state occurs at
glass transition temperature. Liquid-glass transition phenomena are observed
universally in various types of liquids, including molecular liquids, ionic
liquids, metallic liquids, oxides, and chalcogenides [7, 31, 46, 66–68]. There is no
long range order in amorphous materials, however at the liquid-glass transition
a kind of freezing or transition occurs which is similar to that of
second-order phase transformations and which may be possible to characterise
using an order parameter. Amorphous materials (both solid, e.g.,
glasses and liquid, e.g., melts) can be most efficiently studied by
reconstructing structural computer models and analysing coordination
polyhedrons formed by constituent atoms [69]. The general theoretical description of the
topologically disordered glassy state focuses on tessellations [70] and is
based on partitioning space into a set of Voronoi polyhedrons filling the space
of a disordered material. A Voronoi polyhedron is a unit cell around each
structural unit (atom, defect, group of atoms) which contains all the points
closer to this unit than to any other and is an analogue of the Wigner-Seitz
cell in crystals [1–3, 70]. For an
amorphous material, the topological and metric characteristics of the Voronoi
polyhedron of a given unit are defined by its nearest neighbours so that its
structure may be characterised by a distribution of Voronoi polyhedrons. Considerable progress has been achieved in investigating the structure
and distribution of Voronoi polyhedrons of amorphous materials using molecular
dynamics (MD) models [69, 71–73]. MD
simulations reveal that the difference between a liquid and glassy states of an amorphous
material is caused by the formation of percolation clusters in the Voronoi
network, namely, in the liquid state low-density atomic configurations form a
percolation cluster whereas such a percolation cluster does not occur in the
glassy state [71]. The percolation cluster made of low-density atomic configurations
was called a liquid-like cluster as it occurs only in a liquid and does not
occur in the glassy state. Nonetheless, a percolation cluster can be envisaged
in the glassy state but formed by high-density configurations [69, 71].
Solid-like percolation clusters made of high-density configurations seem to exist in all
glass-phase models of spherical atoms and dense spheres [69, 71]. Thus. MD
simulations demonstrate that near Tg  the interconnectivity of atoms
(e.g., the geometry of bonds) changes due to the formation of percolation
clusters composed of coordination Voronoi polyhedrons. While these percolation
clusters made of Voronoi polyhedrons are more mathematical descriptors than
physical objects, their formation results in changes in the derivative
properties of materials near the Tg  [69]. The liquid-glass
transition is thus characterised by a fundamental change in the bond geometries
so that this change can be used to distinguish liquids from glasses although
both have amorphous structures [5, 6, 12, 69]. Many models were proposed to
examine the transition of a liquid to glass at cooling (see overviews in [ 7, 26,
27, 30, 31, 43]). Table 5 outlines basic glass transition models, which are
briefly discussed below.Table 5
Basic glass transition models.
ModelOrdering processKey conceptFree-volumeNoChange
in free (excess) volumeAdam-GibbsNoCooperativity
of motionMode-coupling
theoryNoSelf-trapping
(caging)Kinetically
constrainedNoMobility
defectsFrustrationIcosahedral
ordering in glassy phaseFrustrationThe
Tanaka TOPCrystallisationCompeting
ordering (frustration)Configuron   percolationPercolation
cluster  of broken bonds in liquid phaseBroken
bond (configuron) clustering


### 8.1. Free-Volume Model

The free-volume
model assumes that when a molecule moves by diffusion it has a certain free
volume in its surroundings. The additional (free) volume becomes available
aboveTg in an amount given by (14)Vf=Vf(Tg)+VgΔα(T−Tg), where  Vg  is the molar volume at Tg, Δα is the change in the volume expansion
coefficient which occur at Tg  [60, 74, 75]. The decrease in free
volume while approaching the glass transition temperature gives an explanation
for the increase of viscosity while approaching it. However, pressure
dependence of the viscosity and negative dTg/dP observed for some
liquids are rather difficult to explain by this model and the validity of this
assumption is questioned [27, 43, 76]. Known zero and negative values of Δα are untenable for free volume model as the
free volume contraction could not explain production of relative mobile liquids
above  Tg.


### 8.2. Adam and Gibbs Model

The Adam and Gibbs model
assumes that the lower the temperature is, the larger number of particles
involved in cooperative rearrangements during molecular motion is, for example,
the dynamic coherence lengthξ of molecular motion increases with the decrease
of temperature [61]. The structural relaxation time depends on the
configurational entropy Sconf  as (15)τα=τ0exp(βCSconf), where  τ0  and C are
constants. It is then assumed that Sconf(T)=ΔCp(T−TV)/T,
where ΔCp  is the relaxational part
of the specific heat, TV is the Vogel temperature, which
results in VTF-type equations (see (7)). Although limited by its application, the concept of cooperativity is well describing many
aspects of glass transition [27].


### 8.3. Mode-Coupling Model

The lower the temperature is, the higher the packing
density of an amorphous material is. This leads to
the stronger memory effect via mode couplings, which induces the self-trapping
mechanism in the mode-coupling
theory (MCT). The MCT describes
this self-trapping based on a nonlinear dynamical equation of the density
correlator [30, 77, 78]. The glass transition in MCT
is a purely dynamic transition from an ergodic state which occurs at high temperatures to a
nonergodic state at low temperatures. This transition corresponds to a
bifurcation point of nonlinear equations of motion for density fluctuations
when an infinite cluster of completely caged particles is formed. The
transition from an ergodic to a nonergodic state occurs at the so-called
mode-coupling temperature Tc, which for typical glass formers is  Tc~1.2Tg.  For T<Tc, the
density correlation function develops a nonzero value in the long-time limit (a
finite value of the Edwards-Anderson-order parameter). The MCT describes fast β  relaxation as resulting from rapid local
motion of molecules trapped inside cages, while the slow process of the breakup
of a cage itself leads to the α relaxation. However, analysis shows that there
are no critical singularities above the glass-transition temperature in
contrast to the MCT prediction [27], for example, there is no singular behaviour of viscosity at
Tc. Moreover, there are no heterogeneities in MCT whereas these are
observed experimentally [27]. Trap models which
similarly to MCT regard the glass transition as a dynamic transition consider the
distribution of the waiting time of a particle in a random potential so that
particles are either trapped in cages formed by their neighbours or jump by thermal-activated
rearrangements [27].


### 8.4. The Tanaka Two-Order-Parameter (top) Model

The two-order-parameter (TOP)
model is based on an idea that there are generally two types of local
structures in liquid: normal-liquid structures (NLSs) and locally favoured
structures (LFSs). The liquid is an inhomogeneous-disordered state which has
LFS created and annihilated randomly (in some cases, cooperatively) in a sea of
random NLS. A supercooled liquid which is a frustrated metastable-liquid (the
Griffiths-phase-like) state is in a dynamically heterogeneous state composed of
metastable solid-like islands, which exchange with each other dynamically at
the rate of the structural (α)
relaxation time [79] (Figure 8).Figure 8
Schematic figure
of a supercooled liquid state below melting temperature. LFS: black pentagons,
NLS: grey spheres. Shaded region represents metastable islands with various degrees of
crystal-like order, whose characteristic coherence length isξ. The darker the colour is, the higher the
crystal-like order and the higher the local density are [79]. Courtesy Hajime Tanaka.Actual liquids universally
have a tendency of spontaneous formation of LFS. The liquid-glass transition in
TOP model is controlled by the competition between long-range density ordering
towards crystallisation and short-range bond ordering towards the formation of
LFS due to the incompatibility in their symmetry. Because of this, TOP model
regards vitrification as phenomena that are intrinsically related to crystallisation
in contrast to previous models, which regarded vitrification as a result of a
homogeneous increase in the density and the resulting cooperativity in
molecular motion or the frustration intrinsic to a liquid state itself. TOP
model defines the calorimetric glass-transition temperatureTg as
the temperature where the average lifetime of metastable islands exceeds the
characteristic observation time. The mechanical and volumetric glass-transition
temperature is the temperature where metastable islands, which have a long
enough lifetime comparable to the characteristic observation time, do percolate
[79]. The degree of cooperativity in TOP model is equal to the fraction of
metastable solid-like islands, for example, TOP model operates with two states:
NLS and metastable solid-like islands [79]. The metastable solid-like islands
in TOP model have characteristic nm size ~ξ and are
considered as resulting from random first-order transition. The lifetime of
metastable islands has a wide distribution with the average lifetime equal to
the structural relaxation time. The boson peak corresponds to the localised vibrational
modes characteristic of the LFS and their clusters. The fast β  mode
results from the motion of molecules inside a cage, while the slow β  mode
from the rotational vibrational motion inside a cage within metastable islands.LFSs impede crystallisation because their symmetry is not consistent with
that of the equilibrium crystal. Due to random disorder effects of LFS, a
liquid enters into the Griffiths-phase-like metastable frustrated state below
the melting point,Tm, where its free energy has a complex
multivalley structure, which leads to the non-Arrhenius behaviour of the structural relaxation. The
crossover from a noncooperative to a cooperative regime TOP model describes by
the fraction of the metastable solid-like islands given by the crossover
function (16)f(T)=1exp[κ(T−Tmc)], where κ  controls
the sharpness of transition. The NLS are favoured by density-order parameter, ρ, which increases the local density and leads
to crystallisation, while the LFSs are favoured by bond-order parameter, S¯,
which results from the symmetry-selective parts of the interactions and
increases the quality of bonds. The average fraction of LFS (S¯) is given by (17)S¯≅(gSgρ)exp[β(ΔE−PΔυ)], where β=1/kBT,kB is the
Boltzmann constant, P is the pressure, ΔE and Δυ are the energy gain and the volume change upon
the formation of an LFS, gS and gρ are the degrees of degeneracy of the
states of LFS and NLS, respectively. It is assumed that gS≫gρ  and  ΔE>0.  Δυ can be either positive or negative depending
upon material, for example, Δυ>0 for liquids with tetrahedral units. NLSs
have many possible configurations as well as various bonding states compared
with the unique LFS and there is a large loss of entropy upon the formation of
an LFS, which is given by (18)Δσ=kBln(gSgρ). NLS is a short-lived random structure whereas
LFS a long-lived rigid structural element. The lifetime of an LFS can be
estimated as (19)τLFS=τα0exp(βΔGS), where τα0 is the structural relaxation time of NLS and ΔGS is the energy
barrier to overcome upon the transformation from an LFS to an NLS.TOP model defines fragility
by fraction of LFS: the larger theS¯ is, the stronger (less fragile) the liquid is.
An example of conclusion drawn from TOP model is the increases the fragility of
SiO2 upon the addition of Na2O,  for example, the higher
the Na2O  content is, the higher the fragility which conforms the
experimental data is (see Figure 9).  Na2O acts as a network modifier
breaking the Si–Si links via bridging oxygens. Tanaka proposed that Na2O
destabilises the LFS, for example, that Na2O is the breaker of LFS,
probably, the 6-member ring structures of SiO2. Since Na2O
reduces the number density of LFS (S¯), it increases the fragility of SiO2 and weaken the boson peak [80].Figure 9
Viscosity of two SiO2–Na2O systems.LFSs impede crystallisation
because their symmetry is not consistent with that of the equilibrium crystal.
A similar idea was exploited by Evteev et
al. to explain vitrification of amorphous metals [69, 81]. In addition,
strong liquids should be more difficult to crystallise than fragile below Tg [82].


### 8.5. Frustration Models

Local energetically
preferred structures over simple crystalline packing
impede crystallisation because their symmetry is not consistent with that of
the equilibrium crystal, for example, frustrated over
large distance. Frustration models assume that the glass transition is a
consequence of the geometric frustration [83–86]. Typically, icosahedron is the most compact and stable
from the energy point of view among all coordination polyhedrons encountered in
both ordered and disordered densely packed structures such as amorphous metals.
For example, Kivelson et al. [85, 86] considered
frustration of an icosahedral structure which is the low-symmetry reference
state, into which a liquid tends to be ordered, and ascribed the glass
transition to an avoided critical point of a transition between a liquid and an
ideal icosahedral structure. The Hamiltonian used, for example, in [85, 86] was
similar to that of Steinhardt et al. [84] (20)H=−JS∑i,jS→i⋅S→j+KS∑i≠jS→i⋅S→j|R→i−R→j|, where JS and KS are both positive. The first term which is a short-range
ferromagnetic interaction favours long-range order of the locally preferred
structure, while the second term which is a long-range antiferromagnetic
interaction is responsible for the frustration effects. The ordering is thus
prevented by internal frustration of the order parameter itself as in other
frustration models.Using MD simulations, Evteev et al. [69] showed that in the process of fast cooling of melt iron, the
fraction of atoms for which the coordination polyhedron is an icosahedron (the
Voronoy polyhedron is a dodecahedron) increases most intensely. Moreover,
formation of a percolation cluster from mutually penetrating and contacting
icosahedrons with atoms at vertices and centres provides stabilisation of the
amorphous phase and impedes crystallisation during fast cooling of Fe from melt
[69, 81]. Evteev et al. showed
that glassy phase forms at the glass transition temperature based on a
percolation cluster of mutually penetrating icosahedrons contacting one
another, which contain atoms at the vertices and at the centres (Figure 10). A
fractal cluster of icosahedrons incompatible with translational symmetry plays
the role of binding carcass hampering crystallisation and serves as the
fundamental basis of structural organisation of the glassy (solid amorphous
state) of iron, which basically distinguishes it from the melt.The structure of
amorphous iron resulting from molecular dynamics simulations [69]. (a) The
size of the largest cluster formed by clustered icosahedrons with the
temperature Ti. (b) Projections of the largest cluster
formed by clustered icosahedrons onto one of the faces of the computational
cell at temperatures 1200 K (2) and 1180 K (5). Courtesy Alexander Evteev.
(a)(b)


### 8.6. Kinetically Constrained Models

Kinetically
constrained models consider slow dynamics as of a purely kinetic origin [27, 87],
where dynamical constraints appear below a crossover temperature To or above a corresponding packing fraction, so that above To the
dynamics is liquid-like whereas below To the dynamics becomes
heterogeneous. Hunt, for example, defined the glass as a supercooled liquid,
whose time scale required for equilibration is a percolation relaxation time
and derived the glass transition temperature from equalising the relaxation
time  τc to the experimental time taken arbitrary as texp = 100
seconds  [88]. The Hunt equation (21)Tg≅Em18kB relates the
glass transition temperature Tg with the peak energy in the
distribution of hoping barrier heights Eij of individual relaxation
processes (22)τij=νph−1exp(EijkBT), where νph  is a typical
vibrational frequency roughly 1012 Hz. Using the Coulomb attraction
between opposite charges and Lennard-Jones repulsive interaction, Hunt obtained
an estimation for Em: (23)Em≈qq′4πε0εr0, where q and q′
are ionic charges, r0 is the
equilibrium interionic distance minimising the interaction potential, and ε is
the macroscopic dielectric constant. Accounting that on pressure application
the internal pressure changes to(24)U=−qq′4πε0εr+(qq′48πε0εr0)(r0r)12+P(r3−r03), where
P is pressure, Hunt obtained an excellent description of pressure dependence
of the glass transition temperature Tg(P)  in ionic liquids [76, 89].
Moreover, this approach gave an explanation of reduced glass transition
temperature by confinement in small pores [90]: (25)Tg(∞)−Tg(L)=a(lL), where
a is a constant, l is a typical hopping length, and L is the pore size. In
addition, Hunt explained application of Ehrenfest
theorems to the glass transition [91].


### 8.7. Configuron Percolation Model

In contrast to
other models based on percolation theory, the configuron percolation model of
glass transition considers the percolation not via atoms or bonds, but the
percolation via broken bonds. For example, the configuron percolation model
examines not the transition from the liquid to the glass on decrease of
temperature, but the transition from the glass to liquid on temperature
increase. The transition of a liquid to a glass has many features of
second-order phase transition. Second-order phase transformations are
characterised by symmetry change. The translation-rotation symmetry in the
distribution of atoms and molecules is deemed unchanged at the liquid-glass
transition, which retains the topological disorder of fluids. What kind of
symmetry is changed at glass-liquid transition? To answer to this question, it
is expedient to consider the distribution of configurons (broken bonds) instead
of atoms and to focus the attention on topology of broken bonds at glass-liquid
transition [5, 6, 12, 47]. Consider an ideal disordered
network representing an oxide system such as amorphous SiO2 or  GeO2.
Using the Angell bond lattice model [92], one can represent condensed phases by
their bond network structures. Thus, one can focus the attention on temperature
changes that occur in the system of interconnecting bonds of a disordered
material rather than of atoms. In this approach, the initial set of N strongly
interacting cations such as Si+4 or Ge+4 is replaced by a
congruent set of weakly interacting bonds of the system. The number of bonds
will be Nb=NZ, where Z is the coordination number of cations, for
example, Z = 4 for SiO2 and  GeO2. For amorphous materials
which have no bridging atoms such as amorphous Fe, Si, or Ge, Nb=NZ/2. Figures 11(a) and 11(b) illustrate schematically the replacement of a
disordered atomic structure by the congruent bond structure.Schematic of disordered bond lattice model of an
amorphous material: (a) distribution of atoms in amorphous phase atT=0; (b) distribution of bonds in amorphous
phase at T=0; (c) distribution
of bonds in amorphous phase at T1>0; (d) distribution of bonds in amorphous
phase at higher temperatures T2>T1  when configuron
clustering occurs.
(a)
T=0(b)
T=0(c)
T1>0(d)
T2>T1At
absolute zero temperatureT=0, there are no broken bonds (Figure 11(b)),
however at any finite temperature T, there are thermally activated broken bonds,
for example, configurons (Figure 11(c)). Compared with a crystal lattice of the
same material, the disordered network typically contains significantly more
point defects such as broken bonds or vacancies. For example, the relative
concentration of vacancies in crystalline metals just below the melting point
is only 10−3-10−4 [2, 93]. The energetics of the
disordered net are weaker and point defects can be formed more easily than in
crystals of the same chemical composition. The difference appears from the
thermodynamic parameters of disordered networks. The formation of configurons
is governed by the formation of Gibbs-free energy Gd.
Temperature-induced formation of configurons in a disordered covalent network
can be represented by a reaction involving the breaking of a covalent bond, for
example, in amorphous silica: (26)≡Si•O•Si≡→T≡Si∘O•Si≡. Figure 12 illustrates the
formation of a configuron in the amorphous SiO2. Breaking of a
covalent bond is followed by relaxation effects which lead to the formation of
a relaxed or equilibrium configuron. As
pointed out by Doremus [42, 43], relaxation effects after the breaking of the
bond result in five-coordinated silicon ions.(a) Schematic of 4 covalent bonds (b) and one broken bond in SiO2.
(a)(b)The
higher the temperature is, the higher the concentration of thermally created
configurons is. Because the system of bonds has two states, namely, the ground
state corresponding to unbroken bonds and the excited state corresponding to
broken bonds, it can be described by the statistics of two-level systems and
the two states of the equivalent system are separated by the energy intervalGd  governing (27) [64]. The statistics of two level systems leads to the
well-known relationship for equilibrium concentrations of configurons Cd and unbroken bonds Cu [64, 94] (27)Cd=C0f(T),Cu=C0[1−f(T)],f(T)=exp(−Gd/RT)[1+exp(−Gd/RT)], at absolute zero temperature Cu(0)=C0.
The concentration of configurons gradually increases with the increase of
temperature, and at T→∞  achieves its maximum possible value Cd=0.5C0 if Gd>0. At temperatures close to absolute zero, the
concentration of configurons is very small f(T)→0.
These are homogeneously distributed in the form of single configurons in the
disordered bond network. Configurons motion in the bond network occurs in the
form of thermally activated jumps from site to site and in this case all jump
sites are equivalent in the network. The network, thus, can be characterised as
an ideal 3D-disordered structure which is described by an Euclidean 3D
geometry. As the temperature increases, the concentration of configurons
gradually increases. The higher the temperature is, the higher the
concentration of configurons and, hence, some of them inevitably will be in the
vicinity of others. Two and more nearby configurons form clusters of
configurons and the higher the concentration of configurons is, the higher the
probability of their clustering is (Figure 11(d)). Although configuron clusters
are dynamic structures, the higher the temperature is, the larger are clusters
made of configurons in the disordered bond network. As known from the percolation
theory, when the concentration of configurons exceeds the threshold level, they will form the macroscopic
so-called percolation cluster, which penetrates the whole volume of the
disordered network [95]. The percolation cluster made of broken bonds forms
at glass transition temperature and grows in size with the increase of
temperature.The configurons
are moving in the disordered network, therefore, the percolation cluster made
of broken bonds is a dynamic structure which changes its configuration
remaining, however, an infinite percolation cluster. The percolation cluster is
made entirely of broken bonds and hence is readily available for a more
percolative than a site-to-site diffusive motion of configurons. Hence, above
the percolation level the motion of configurons in the bond network occurs via
preferred pathways through the percolation cluster. The percolation cluster near
the percolation threshold is fractal in dimension, therefore the bond system of an amorphous
material changes its Hausdorff-Besikovitch dimensionality from Euclidian 3
below theTg (where the amorphous material is solid), to fractal
2.55±0.05 above the Tg, where the amorphous material is liquid [5, 6, 12].As
the bond network of an amorphous material is disordered, the concentration of
configurons at which the percolation threshold is achieved can be found using
the universal critical percolation densityfc,
which remains the same for both ordered and disordered lattices [95, 96]. The
relative concentration of broken bonds is given by f(T)=Cd/C0 which shows that the higher the temperature
is, the higher is f(T).
Assuming that at Cd/C0=1 the space is completely filled by configurons,
one can designate f(T) as the volume fraction of space occupied by
configurons. Thus, the critical (glass-transition) temperature Tg at
which the percolation threshold is achieved can be found assuming that the
configurons achieve the universal critical density given by the percolation
theory (28)f(Tg)=fc. For
SiO2 and GeO2, it was supposed that fc=ϑc,
where ϑc is the Scher-Zallen critical density in the 3D
space ϑc=0.15±0.01 [95–97]. For many
percolating systems, the value of fc is significantly lower compared to ϑc [96].At
temperatures aboveTg, the space is filled by configurons at
concentrations which exceed the critical density fc,
therefore they form the percolation cluster with fractal geometry changing the
state of material from solid-like (glass) to liquid-like. This leads to the
following equation of glass transition temperature [5, 6, 12]: (29)Tg=Hd[Sd+Rln[(1−fc)/fc]]. Note that because Sd≫R, this
equation can be simplified to Tg≈Hd/Sd, which is an
analogue of the Diennes ratio used to assess the melting temperatures of
crystalline solids. Below the Tg, the configurons are uniformly
distributed in space, and formation of clusters is improbable. The geometry of
network defects in this area can be characterised as 3D Euclidean. With the increase
of temperature at T=Tg, the concentration of defects achieves the
critical concentration for formation of a percolation cluster. Above the Tg,
a percolation cluster made of configurons is formed, and the geometry of the
network becomes fractal with the fractal dimension 2.55±0.05 near the Tg.Equation (29) gives excellent data for glass-transition
temperatures [5, 6, 12, 47]. Note that the glass-transition temperature (29)
depends on thermal history for several reasons: (i) during cooling, a part of
material is inevitably crystallised, (ii) configurons need a certain time to
relax to their equilibrium sizes, and (iii) the enthalpy of configuron
formation depends on the overall state of amorphous material including its
quenched density which can be assessed from [62]  Hd≈q1q2e2/εdaN,
where qz  is the valence of
the cation-anion pair, e is the
standard unit of charge, da  is the average bond distance, N is
the coordination number, and ε is the
dielectric constant of the glass which depends on the thermal history. This
estimation is similar to the Hunt assessment (24)  [89].The
characteristic linear scale which describes the branch sizes of dynamic clusters
formed by configurons is the correlation lengthξ(T). Below the Tg, the correlation
length gives characteristic sizes of clusters made of configurons whereas above the Tg,
it shows characteristic sizes of clusters made of unbroken bonds, for example,
atoms. ξ(T) gives
the linear dimension above which the material is homogeneous, for example, a
material with sizes larger than ξ(T) has
on average uniformly distributed configurons. At sizes smaller compared to ξ(T), the amorphous material is dynamically inhomogeneous,
for example, has a fractal geometry [95, 96]. Because of the formation of
percolation cluster, the material has a fractal geometry at lengths smaller
than ξ(T). It
means that the glass loses
at glass-liquid transition the invariance for the Euclidian space isometries
such as translation and rotation on length scales smaller than ξ(T). The liquid near the glass transition is
dynamically inhomogeneous on length scales smaller than ξ(T) and remains unchanged for fractal space
group of isometries. Figure 13 shows schematically the structure of an
amorphous material near the glass transition temperature.Figure 13
Schematic representation of the dynamic homogeneous
fractal structure of an amorphous material near the glass transition
temperature. Below theTg, the correlation length gives
characteristic sizes of clusters made of configurons, whereas above the Tg,
it shows characteristic sizes of clusters made of unbroken bonds. The higher
the temperature is, the smaller sizes of fractal volumes are. Note that the
structure shown is dynamical, for example, changes with time due to configuron
diffusion.The
fractal dimension of percolation clusters near the percolation threshold isdf=2.55±0.05. At temperatures far from Tg, the correlation length is
small, whereas at temperatures approaching Tg, it diverges (30)ξ(T)=ξ0|f(T)−fc|ν, where
the critical exponent ν = 0.88  [93, 94].If
sample sizes are smaller thanξ(T), the
amorphous material is dynamically inhomogeneous and has a fractal geometry. Finite
size effects in the glass transition are described as a drift to lower values
of   Tg; when sample sizes L  decrease [6], (31)Tg(∞)−Tg(L)=0.1275Tg(RTgHd)(ξ0L)1.136, one can
see that this expression conforms well with (25), for example, with the Hunt
results [90]. The heat capacity per mole of configurons involved in the
percolation cluster near  Tg  was found as [6, 12]  (32)Cp,conf=R(HdRT)2f(T)[1−f(T)]×(1+βP0(ΔH/Hd)T11−β|T−Tg|(1−β)), where T1=RTg2/θc(1−θc)Hd, β =  0.41 is the critical exponent [95, 96], P0 is a numerical coefficient close to unity, for example, for strong liquids P0 = 1.0695,
and ΔH≪Hd  is the enthalpy of configurons in the percolation cluster.The
configuron model of glass transition shows that the linear expansion
coefficient near theTg behaves as [6] (33)αconf=Nb(ΔV0HdVRT2)f(T)[1−f(T)]×(1+βP0(ΔH/Hd)T11−β|T−Tg|(1−β)), where
ΔV0/V
is the relative change of volume per one broken bond. One can observe, hence,
that both thermal expansion coefficient and heat capacity show divergences near
Tg  proportional to ∝1/|T−Tg|0.59  [6, 12] (see Figure 1).Complex
oxide systems are typically fragile. These are described by a modified random
network model comprising network modifying cations distributed in channels, and
the value offc in these systems is significantly lower
compared to strong materials as can be seen from Table 6 [47].Table 6
Glass transition temperatures of amorphous
materials.
Amorphous materialRDTg/KfcSilica
(SiO2)1.4514750.15Germania (GeO2)1.477860.15SLS
(mass%):2.168701.58×10−370SiO2 21CaO 9Na2OB2O33.285809.14×10−5Diopside
(CaMgSi2O6)4.519786.35×10−7Anorthite
(CaAl2Si2O8)4.5211263.38×10−7Data
from Table6 show that the higher the fragility ratio is, the lower the
threshold for the formation of percolation clusters of configurons in the
material is. There is a direct anticorrelation between the fragility ratio and
configuron percolation threshold which determines the glass transition
temperature. Networks that exhibit only small changes in the activation energy
for flow with temperature form percolation clusters of configurons at the
classical Scher-Zallen critical density. In contrast, fragile liquids, which
are characterised by a higher density of configurational states, have a very
low percolation threshold which decreases with increasing fragility.Angell interpreted strong and
fragile behaviours
of liquids in terms
of differences in topology of the configuration space potential energy
hypersurfaces [55]; for example, fragile liquids have a higher density of configurational
states and hence a higher degeneracy leading to rapid thermal excitations. In
the bond lattice model of amorphous materials, the system of strongly interacting ions is
replaced by a congruent set of weakly interacting bonds. The glass transition is related in this model with formation
of percolation clusters made of configurons and change of bond
Hausdorff-Besikovitch dimension [5, 6]. The
diminishing values of configuron percolation thresholds can be interpreted in terms of configuron
size or delocalisation. It is deemed that in fragile liquids the configurons
are larger, for example, delocalised; moreover, the higher the fragility ratio
is, the larger the effective configuron radius and its delocalisation are. Due
to configuron delocalisation, the glass transition, which is associated with
the increase of bond Hausdorff-Besikovitch dimension from df=2.55±0.05 to 3, occurs in fragile liquids at lower percolation threshold compared to
strong liquids. The effective configuron radius, rc,
can be assessed from (34)rc=rd(ϑcfc)1/3, where rd is the bond radius (half of bond length). For
strong liquids, fc=ϑc and thus the configuron radii are equal to
bond radii. Strong materials
such as silica have small radii configurons localised on broken bonds and
because of that they should show a smaller dependence on thermal history which conforms
to experimental findings [98]. In fragile materials, the effective configuron radii considerably exceed bond
radii, rc≫rd.
For example, B2O3 with fragility ratio RD = 3.28
has rc=11.79rd,
which is due to its specific structure. Both crystalline and vitreous boron
oxides consist of planar oxygen triangles centred by boron most of which
accordingly to X-ray diffraction data are arranged in boroxol rings (see [7]).
The two-dimensional nature of the B2O3 network means that
the third direction is added by crumbling of the planar structures in a
three-dimensional amorphous boric oxide which result in effective large size
configuron compared to bond length.


## 8.1. Free-Volume Model

The free-volume
model assumes that when a molecule moves by diffusion it has a certain free
volume in its surroundings. The additional (free) volume becomes available
aboveTg in an amount given by (14)Vf=Vf(Tg)+VgΔα(T−Tg), where  Vg  is the molar volume at Tg, Δα is the change in the volume expansion
coefficient which occur at Tg  [60, 74, 75]. The decrease in free
volume while approaching the glass transition temperature gives an explanation
for the increase of viscosity while approaching it. However, pressure
dependence of the viscosity and negative dTg/dP observed for some
liquids are rather difficult to explain by this model and the validity of this
assumption is questioned [27, 43, 76]. Known zero and negative values of Δα are untenable for free volume model as the
free volume contraction could not explain production of relative mobile liquids
above  Tg.


## 8.2. Adam and Gibbs Model

The Adam and Gibbs model
assumes that the lower the temperature is, the larger number of particles
involved in cooperative rearrangements during molecular motion is, for example,
the dynamic coherence lengthξ of molecular motion increases with the decrease
of temperature [61]. The structural relaxation time depends on the
configurational entropy Sconf  as (15)τα=τ0exp(βCSconf), where  τ0  and C are
constants. It is then assumed that Sconf(T)=ΔCp(T−TV)/T,
where ΔCp  is the relaxational part
of the specific heat, TV is the Vogel temperature, which
results in VTF-type equations (see (7)). Although limited by its application, the concept of cooperativity is well describing many
aspects of glass transition [27].


## 8.3. Mode-Coupling Model

The lower the temperature is, the higher the packing
density of an amorphous material is. This leads to
the stronger memory effect via mode couplings, which induces the self-trapping
mechanism in the mode-coupling
theory (MCT). The MCT describes
this self-trapping based on a nonlinear dynamical equation of the density
correlator [30, 77, 78]. The glass transition in MCT
is a purely dynamic transition from an ergodic state which occurs at high temperatures to a
nonergodic state at low temperatures. This transition corresponds to a
bifurcation point of nonlinear equations of motion for density fluctuations
when an infinite cluster of completely caged particles is formed. The
transition from an ergodic to a nonergodic state occurs at the so-called
mode-coupling temperature Tc, which for typical glass formers is  Tc~1.2Tg.  For T<Tc, the
density correlation function develops a nonzero value in the long-time limit (a
finite value of the Edwards-Anderson-order parameter). The MCT describes fast β  relaxation as resulting from rapid local
motion of molecules trapped inside cages, while the slow process of the breakup
of a cage itself leads to the α relaxation. However, analysis shows that there
are no critical singularities above the glass-transition temperature in
contrast to the MCT prediction [27], for example, there is no singular behaviour of viscosity at
Tc. Moreover, there are no heterogeneities in MCT whereas these are
observed experimentally [27]. Trap models which
similarly to MCT regard the glass transition as a dynamic transition consider the
distribution of the waiting time of a particle in a random potential so that
particles are either trapped in cages formed by their neighbours or jump by thermal-activated
rearrangements [27].


## 8.4. The Tanaka Two-Order-Parameter (top) Model

The two-order-parameter (TOP)
model is based on an idea that there are generally two types of local
structures in liquid: normal-liquid structures (NLSs) and locally favoured
structures (LFSs). The liquid is an inhomogeneous-disordered state which has
LFS created and annihilated randomly (in some cases, cooperatively) in a sea of
random NLS. A supercooled liquid which is a frustrated metastable-liquid (the
Griffiths-phase-like) state is in a dynamically heterogeneous state composed of
metastable solid-like islands, which exchange with each other dynamically at
the rate of the structural (α)
relaxation time [79] (Figure 8).Figure 8
Schematic figure
of a supercooled liquid state below melting temperature. LFS: black pentagons,
NLS: grey spheres. Shaded region represents metastable islands with various degrees of
crystal-like order, whose characteristic coherence length isξ. The darker the colour is, the higher the
crystal-like order and the higher the local density are [79]. Courtesy Hajime Tanaka.Actual liquids universally
have a tendency of spontaneous formation of LFS. The liquid-glass transition in
TOP model is controlled by the competition between long-range density ordering
towards crystallisation and short-range bond ordering towards the formation of
LFS due to the incompatibility in their symmetry. Because of this, TOP model
regards vitrification as phenomena that are intrinsically related to crystallisation
in contrast to previous models, which regarded vitrification as a result of a
homogeneous increase in the density and the resulting cooperativity in
molecular motion or the frustration intrinsic to a liquid state itself. TOP
model defines the calorimetric glass-transition temperatureTg as
the temperature where the average lifetime of metastable islands exceeds the
characteristic observation time. The mechanical and volumetric glass-transition
temperature is the temperature where metastable islands, which have a long
enough lifetime comparable to the characteristic observation time, do percolate
[79]. The degree of cooperativity in TOP model is equal to the fraction of
metastable solid-like islands, for example, TOP model operates with two states:
NLS and metastable solid-like islands [79]. The metastable solid-like islands
in TOP model have characteristic nm size ~ξ and are
considered as resulting from random first-order transition. The lifetime of
metastable islands has a wide distribution with the average lifetime equal to
the structural relaxation time. The boson peak corresponds to the localised vibrational
modes characteristic of the LFS and their clusters. The fast β  mode
results from the motion of molecules inside a cage, while the slow β  mode
from the rotational vibrational motion inside a cage within metastable islands.LFSs impede crystallisation because their symmetry is not consistent with
that of the equilibrium crystal. Due to random disorder effects of LFS, a
liquid enters into the Griffiths-phase-like metastable frustrated state below
the melting point,Tm, where its free energy has a complex
multivalley structure, which leads to the non-Arrhenius behaviour of the structural relaxation. The
crossover from a noncooperative to a cooperative regime TOP model describes by
the fraction of the metastable solid-like islands given by the crossover
function (16)f(T)=1exp[κ(T−Tmc)], where κ  controls
the sharpness of transition. The NLS are favoured by density-order parameter, ρ, which increases the local density and leads
to crystallisation, while the LFSs are favoured by bond-order parameter, S¯,
which results from the symmetry-selective parts of the interactions and
increases the quality of bonds. The average fraction of LFS (S¯) is given by (17)S¯≅(gSgρ)exp[β(ΔE−PΔυ)], where β=1/kBT,kB is the
Boltzmann constant, P is the pressure, ΔE and Δυ are the energy gain and the volume change upon
the formation of an LFS, gS and gρ are the degrees of degeneracy of the
states of LFS and NLS, respectively. It is assumed that gS≫gρ  and  ΔE>0.  Δυ can be either positive or negative depending
upon material, for example, Δυ>0 for liquids with tetrahedral units. NLSs
have many possible configurations as well as various bonding states compared
with the unique LFS and there is a large loss of entropy upon the formation of
an LFS, which is given by (18)Δσ=kBln(gSgρ). NLS is a short-lived random structure whereas
LFS a long-lived rigid structural element. The lifetime of an LFS can be
estimated as (19)τLFS=τα0exp(βΔGS), where τα0 is the structural relaxation time of NLS and ΔGS is the energy
barrier to overcome upon the transformation from an LFS to an NLS.TOP model defines fragility
by fraction of LFS: the larger theS¯ is, the stronger (less fragile) the liquid is.
An example of conclusion drawn from TOP model is the increases the fragility of
SiO2 upon the addition of Na2O,  for example, the higher
the Na2O  content is, the higher the fragility which conforms the
experimental data is (see Figure 9).  Na2O acts as a network modifier
breaking the Si–Si links via bridging oxygens. Tanaka proposed that Na2O
destabilises the LFS, for example, that Na2O is the breaker of LFS,
probably, the 6-member ring structures of SiO2. Since Na2O
reduces the number density of LFS (S¯), it increases the fragility of SiO2 and weaken the boson peak [80].Figure 9
Viscosity of two SiO2–Na2O systems.LFSs impede crystallisation
because their symmetry is not consistent with that of the equilibrium crystal.
A similar idea was exploited by Evteev et
al. to explain vitrification of amorphous metals [69, 81]. In addition,
strong liquids should be more difficult to crystallise than fragile below Tg [82].


## 8.5. Frustration Models

Local energetically
preferred structures over simple crystalline packing
impede crystallisation because their symmetry is not consistent with that of
the equilibrium crystal, for example, frustrated over
large distance. Frustration models assume that the glass transition is a
consequence of the geometric frustration [83–86]. Typically, icosahedron is the most compact and stable
from the energy point of view among all coordination polyhedrons encountered in
both ordered and disordered densely packed structures such as amorphous metals.
For example, Kivelson et al. [85, 86] considered
frustration of an icosahedral structure which is the low-symmetry reference
state, into which a liquid tends to be ordered, and ascribed the glass
transition to an avoided critical point of a transition between a liquid and an
ideal icosahedral structure. The Hamiltonian used, for example, in [85, 86] was
similar to that of Steinhardt et al. [84] (20)H=−JS∑i,jS→i⋅S→j+KS∑i≠jS→i⋅S→j|R→i−R→j|, where JS and KS are both positive. The first term which is a short-range
ferromagnetic interaction favours long-range order of the locally preferred
structure, while the second term which is a long-range antiferromagnetic
interaction is responsible for the frustration effects. The ordering is thus
prevented by internal frustration of the order parameter itself as in other
frustration models.Using MD simulations, Evteev et al. [69] showed that in the process of fast cooling of melt iron, the
fraction of atoms for which the coordination polyhedron is an icosahedron (the
Voronoy polyhedron is a dodecahedron) increases most intensely. Moreover,
formation of a percolation cluster from mutually penetrating and contacting
icosahedrons with atoms at vertices and centres provides stabilisation of the
amorphous phase and impedes crystallisation during fast cooling of Fe from melt
[69, 81]. Evteev et al. showed
that glassy phase forms at the glass transition temperature based on a
percolation cluster of mutually penetrating icosahedrons contacting one
another, which contain atoms at the vertices and at the centres (Figure 10). A
fractal cluster of icosahedrons incompatible with translational symmetry plays
the role of binding carcass hampering crystallisation and serves as the
fundamental basis of structural organisation of the glassy (solid amorphous
state) of iron, which basically distinguishes it from the melt.The structure of
amorphous iron resulting from molecular dynamics simulations [69]. (a) The
size of the largest cluster formed by clustered icosahedrons with the
temperature Ti. (b) Projections of the largest cluster
formed by clustered icosahedrons onto one of the faces of the computational
cell at temperatures 1200 K (2) and 1180 K (5). Courtesy Alexander Evteev.
(a)(b)


## 8.6. Kinetically Constrained Models

Kinetically
constrained models consider slow dynamics as of a purely kinetic origin [27, 87],
where dynamical constraints appear below a crossover temperature To or above a corresponding packing fraction, so that above To the
dynamics is liquid-like whereas below To the dynamics becomes
heterogeneous. Hunt, for example, defined the glass as a supercooled liquid,
whose time scale required for equilibration is a percolation relaxation time
and derived the glass transition temperature from equalising the relaxation
time  τc to the experimental time taken arbitrary as texp = 100
seconds  [88]. The Hunt equation (21)Tg≅Em18kB relates the
glass transition temperature Tg with the peak energy in the
distribution of hoping barrier heights Eij of individual relaxation
processes (22)τij=νph−1exp(EijkBT), where νph  is a typical
vibrational frequency roughly 1012 Hz. Using the Coulomb attraction
between opposite charges and Lennard-Jones repulsive interaction, Hunt obtained
an estimation for Em: (23)Em≈qq′4πε0εr0, where q and q′
are ionic charges, r0 is the
equilibrium interionic distance minimising the interaction potential, and ε is
the macroscopic dielectric constant. Accounting that on pressure application
the internal pressure changes to(24)U=−qq′4πε0εr+(qq′48πε0εr0)(r0r)12+P(r3−r03), where
P is pressure, Hunt obtained an excellent description of pressure dependence
of the glass transition temperature Tg(P)  in ionic liquids [76, 89].
Moreover, this approach gave an explanation of reduced glass transition
temperature by confinement in small pores [90]: (25)Tg(∞)−Tg(L)=a(lL), where
a is a constant, l is a typical hopping length, and L is the pore size. In
addition, Hunt explained application of Ehrenfest
theorems to the glass transition [91].


## 8.7. Configuron Percolation Model

In contrast to
other models based on percolation theory, the configuron percolation model of
glass transition considers the percolation not via atoms or bonds, but the
percolation via broken bonds. For example, the configuron percolation model
examines not the transition from the liquid to the glass on decrease of
temperature, but the transition from the glass to liquid on temperature
increase. The transition of a liquid to a glass has many features of
second-order phase transition. Second-order phase transformations are
characterised by symmetry change. The translation-rotation symmetry in the
distribution of atoms and molecules is deemed unchanged at the liquid-glass
transition, which retains the topological disorder of fluids. What kind of
symmetry is changed at glass-liquid transition? To answer to this question, it
is expedient to consider the distribution of configurons (broken bonds) instead
of atoms and to focus the attention on topology of broken bonds at glass-liquid
transition [5, 6, 12, 47]. Consider an ideal disordered
network representing an oxide system such as amorphous SiO2 or  GeO2.
Using the Angell bond lattice model [92], one can represent condensed phases by
their bond network structures. Thus, one can focus the attention on temperature
changes that occur in the system of interconnecting bonds of a disordered
material rather than of atoms. In this approach, the initial set of N strongly
interacting cations such as Si+4 or Ge+4 is replaced by a
congruent set of weakly interacting bonds of the system. The number of bonds
will be Nb=NZ, where Z is the coordination number of cations, for
example, Z = 4 for SiO2 and  GeO2. For amorphous materials
which have no bridging atoms such as amorphous Fe, Si, or Ge, Nb=NZ/2. Figures 11(a) and 11(b) illustrate schematically the replacement of a
disordered atomic structure by the congruent bond structure.Schematic of disordered bond lattice model of an
amorphous material: (a) distribution of atoms in amorphous phase atT=0; (b) distribution of bonds in amorphous
phase at T=0; (c) distribution
of bonds in amorphous phase at T1>0; (d) distribution of bonds in amorphous
phase at higher temperatures T2>T1  when configuron
clustering occurs.
(a)
T=0(b)
T=0(c)
T1>0(d)
T2>T1At
absolute zero temperatureT=0, there are no broken bonds (Figure 11(b)),
however at any finite temperature T, there are thermally activated broken bonds,
for example, configurons (Figure 11(c)). Compared with a crystal lattice of the
same material, the disordered network typically contains significantly more
point defects such as broken bonds or vacancies. For example, the relative
concentration of vacancies in crystalline metals just below the melting point
is only 10−3-10−4 [2, 93]. The energetics of the
disordered net are weaker and point defects can be formed more easily than in
crystals of the same chemical composition. The difference appears from the
thermodynamic parameters of disordered networks. The formation of configurons
is governed by the formation of Gibbs-free energy Gd.
Temperature-induced formation of configurons in a disordered covalent network
can be represented by a reaction involving the breaking of a covalent bond, for
example, in amorphous silica: (26)≡Si•O•Si≡→T≡Si∘O•Si≡. Figure 12 illustrates the
formation of a configuron in the amorphous SiO2. Breaking of a
covalent bond is followed by relaxation effects which lead to the formation of
a relaxed or equilibrium configuron. As
pointed out by Doremus [42, 43], relaxation effects after the breaking of the
bond result in five-coordinated silicon ions.(a) Schematic of 4 covalent bonds (b) and one broken bond in SiO2.
(a)(b)The
higher the temperature is, the higher the concentration of thermally created
configurons is. Because the system of bonds has two states, namely, the ground
state corresponding to unbroken bonds and the excited state corresponding to
broken bonds, it can be described by the statistics of two-level systems and
the two states of the equivalent system are separated by the energy intervalGd  governing (27) [64]. The statistics of two level systems leads to the
well-known relationship for equilibrium concentrations of configurons Cd and unbroken bonds Cu [64, 94] (27)Cd=C0f(T),Cu=C0[1−f(T)],f(T)=exp(−Gd/RT)[1+exp(−Gd/RT)], at absolute zero temperature Cu(0)=C0.
The concentration of configurons gradually increases with the increase of
temperature, and at T→∞  achieves its maximum possible value Cd=0.5C0 if Gd>0. At temperatures close to absolute zero, the
concentration of configurons is very small f(T)→0.
These are homogeneously distributed in the form of single configurons in the
disordered bond network. Configurons motion in the bond network occurs in the
form of thermally activated jumps from site to site and in this case all jump
sites are equivalent in the network. The network, thus, can be characterised as
an ideal 3D-disordered structure which is described by an Euclidean 3D
geometry. As the temperature increases, the concentration of configurons
gradually increases. The higher the temperature is, the higher the
concentration of configurons and, hence, some of them inevitably will be in the
vicinity of others. Two and more nearby configurons form clusters of
configurons and the higher the concentration of configurons is, the higher the
probability of their clustering is (Figure 11(d)). Although configuron clusters
are dynamic structures, the higher the temperature is, the larger are clusters
made of configurons in the disordered bond network. As known from the percolation
theory, when the concentration of configurons exceeds the threshold level, they will form the macroscopic
so-called percolation cluster, which penetrates the whole volume of the
disordered network [95]. The percolation cluster made of broken bonds forms
at glass transition temperature and grows in size with the increase of
temperature.The configurons
are moving in the disordered network, therefore, the percolation cluster made
of broken bonds is a dynamic structure which changes its configuration
remaining, however, an infinite percolation cluster. The percolation cluster is
made entirely of broken bonds and hence is readily available for a more
percolative than a site-to-site diffusive motion of configurons. Hence, above
the percolation level the motion of configurons in the bond network occurs via
preferred pathways through the percolation cluster. The percolation cluster near
the percolation threshold is fractal in dimension, therefore the bond system of an amorphous
material changes its Hausdorff-Besikovitch dimensionality from Euclidian 3
below theTg (where the amorphous material is solid), to fractal
2.55±0.05 above the Tg, where the amorphous material is liquid [5, 6, 12].As
the bond network of an amorphous material is disordered, the concentration of
configurons at which the percolation threshold is achieved can be found using
the universal critical percolation densityfc,
which remains the same for both ordered and disordered lattices [95, 96]. The
relative concentration of broken bonds is given by f(T)=Cd/C0 which shows that the higher the temperature
is, the higher is f(T).
Assuming that at Cd/C0=1 the space is completely filled by configurons,
one can designate f(T) as the volume fraction of space occupied by
configurons. Thus, the critical (glass-transition) temperature Tg at
which the percolation threshold is achieved can be found assuming that the
configurons achieve the universal critical density given by the percolation
theory (28)f(Tg)=fc. For
SiO2 and GeO2, it was supposed that fc=ϑc,
where ϑc is the Scher-Zallen critical density in the 3D
space ϑc=0.15±0.01 [95–97]. For many
percolating systems, the value of fc is significantly lower compared to ϑc [96].At
temperatures aboveTg, the space is filled by configurons at
concentrations which exceed the critical density fc,
therefore they form the percolation cluster with fractal geometry changing the
state of material from solid-like (glass) to liquid-like. This leads to the
following equation of glass transition temperature [5, 6, 12]: (29)Tg=Hd[Sd+Rln[(1−fc)/fc]]. Note that because Sd≫R, this
equation can be simplified to Tg≈Hd/Sd, which is an
analogue of the Diennes ratio used to assess the melting temperatures of
crystalline solids. Below the Tg, the configurons are uniformly
distributed in space, and formation of clusters is improbable. The geometry of
network defects in this area can be characterised as 3D Euclidean. With the increase
of temperature at T=Tg, the concentration of defects achieves the
critical concentration for formation of a percolation cluster. Above the Tg,
a percolation cluster made of configurons is formed, and the geometry of the
network becomes fractal with the fractal dimension 2.55±0.05 near the Tg.Equation (29) gives excellent data for glass-transition
temperatures [5, 6, 12, 47]. Note that the glass-transition temperature (29)
depends on thermal history for several reasons: (i) during cooling, a part of
material is inevitably crystallised, (ii) configurons need a certain time to
relax to their equilibrium sizes, and (iii) the enthalpy of configuron
formation depends on the overall state of amorphous material including its
quenched density which can be assessed from [62]  Hd≈q1q2e2/εdaN,
where qz  is the valence of
the cation-anion pair, e is the
standard unit of charge, da  is the average bond distance, N is
the coordination number, and ε is the
dielectric constant of the glass which depends on the thermal history. This
estimation is similar to the Hunt assessment (24)  [89].The
characteristic linear scale which describes the branch sizes of dynamic clusters
formed by configurons is the correlation lengthξ(T). Below the Tg, the correlation
length gives characteristic sizes of clusters made of configurons whereas above the Tg,
it shows characteristic sizes of clusters made of unbroken bonds, for example,
atoms. ξ(T) gives
the linear dimension above which the material is homogeneous, for example, a
material with sizes larger than ξ(T) has
on average uniformly distributed configurons. At sizes smaller compared to ξ(T), the amorphous material is dynamically inhomogeneous,
for example, has a fractal geometry [95, 96]. Because of the formation of
percolation cluster, the material has a fractal geometry at lengths smaller
than ξ(T). It
means that the glass loses
at glass-liquid transition the invariance for the Euclidian space isometries
such as translation and rotation on length scales smaller than ξ(T). The liquid near the glass transition is
dynamically inhomogeneous on length scales smaller than ξ(T) and remains unchanged for fractal space
group of isometries. Figure 13 shows schematically the structure of an
amorphous material near the glass transition temperature.Figure 13
Schematic representation of the dynamic homogeneous
fractal structure of an amorphous material near the glass transition
temperature. Below theTg, the correlation length gives
characteristic sizes of clusters made of configurons, whereas above the Tg,
it shows characteristic sizes of clusters made of unbroken bonds. The higher
the temperature is, the smaller sizes of fractal volumes are. Note that the
structure shown is dynamical, for example, changes with time due to configuron
diffusion.The
fractal dimension of percolation clusters near the percolation threshold isdf=2.55±0.05. At temperatures far from Tg, the correlation length is
small, whereas at temperatures approaching Tg, it diverges (30)ξ(T)=ξ0|f(T)−fc|ν, where
the critical exponent ν = 0.88  [93, 94].If
sample sizes are smaller thanξ(T), the
amorphous material is dynamically inhomogeneous and has a fractal geometry. Finite
size effects in the glass transition are described as a drift to lower values
of   Tg; when sample sizes L  decrease [6], (31)Tg(∞)−Tg(L)=0.1275Tg(RTgHd)(ξ0L)1.136, one can
see that this expression conforms well with (25), for example, with the Hunt
results [90]. The heat capacity per mole of configurons involved in the
percolation cluster near  Tg  was found as [6, 12]  (32)Cp,conf=R(HdRT)2f(T)[1−f(T)]×(1+βP0(ΔH/Hd)T11−β|T−Tg|(1−β)), where T1=RTg2/θc(1−θc)Hd, β =  0.41 is the critical exponent [95, 96], P0 is a numerical coefficient close to unity, for example, for strong liquids P0 = 1.0695,
and ΔH≪Hd  is the enthalpy of configurons in the percolation cluster.The
configuron model of glass transition shows that the linear expansion
coefficient near theTg behaves as [6] (33)αconf=Nb(ΔV0HdVRT2)f(T)[1−f(T)]×(1+βP0(ΔH/Hd)T11−β|T−Tg|(1−β)), where
ΔV0/V
is the relative change of volume per one broken bond. One can observe, hence,
that both thermal expansion coefficient and heat capacity show divergences near
Tg  proportional to ∝1/|T−Tg|0.59  [6, 12] (see Figure 1).Complex
oxide systems are typically fragile. These are described by a modified random
network model comprising network modifying cations distributed in channels, and
the value offc in these systems is significantly lower
compared to strong materials as can be seen from Table 6 [47].Table 6
Glass transition temperatures of amorphous
materials.
Amorphous materialRDTg/KfcSilica
(SiO2)1.4514750.15Germania (GeO2)1.477860.15SLS
(mass%):2.168701.58×10−370SiO2 21CaO 9Na2OB2O33.285809.14×10−5Diopside
(CaMgSi2O6)4.519786.35×10−7Anorthite
(CaAl2Si2O8)4.5211263.38×10−7Data
from Table6 show that the higher the fragility ratio is, the lower the
threshold for the formation of percolation clusters of configurons in the
material is. There is a direct anticorrelation between the fragility ratio and
configuron percolation threshold which determines the glass transition
temperature. Networks that exhibit only small changes in the activation energy
for flow with temperature form percolation clusters of configurons at the
classical Scher-Zallen critical density. In contrast, fragile liquids, which
are characterised by a higher density of configurational states, have a very
low percolation threshold which decreases with increasing fragility.Angell interpreted strong and
fragile behaviours
of liquids in terms
of differences in topology of the configuration space potential energy
hypersurfaces [55]; for example, fragile liquids have a higher density of configurational
states and hence a higher degeneracy leading to rapid thermal excitations. In
the bond lattice model of amorphous materials, the system of strongly interacting ions is
replaced by a congruent set of weakly interacting bonds. The glass transition is related in this model with formation
of percolation clusters made of configurons and change of bond
Hausdorff-Besikovitch dimension [5, 6]. The
diminishing values of configuron percolation thresholds can be interpreted in terms of configuron
size or delocalisation. It is deemed that in fragile liquids the configurons
are larger, for example, delocalised; moreover, the higher the fragility ratio
is, the larger the effective configuron radius and its delocalisation are. Due
to configuron delocalisation, the glass transition, which is associated with
the increase of bond Hausdorff-Besikovitch dimension from df=2.55±0.05 to 3, occurs in fragile liquids at lower percolation threshold compared to
strong liquids. The effective configuron radius, rc,
can be assessed from (34)rc=rd(ϑcfc)1/3, where rd is the bond radius (half of bond length). For
strong liquids, fc=ϑc and thus the configuron radii are equal to
bond radii. Strong materials
such as silica have small radii configurons localised on broken bonds and
because of that they should show a smaller dependence on thermal history which conforms
to experimental findings [98]. In fragile materials, the effective configuron radii considerably exceed bond
radii, rc≫rd.
For example, B2O3 with fragility ratio RD = 3.28
has rc=11.79rd,
which is due to its specific structure. Both crystalline and vitreous boron
oxides consist of planar oxygen triangles centred by boron most of which
accordingly to X-ray diffraction data are arranged in boroxol rings (see [7]).
The two-dimensional nature of the B2O3 network means that
the third direction is added by crumbling of the planar structures in a
three-dimensional amorphous boric oxide which result in effective large size
configuron compared to bond length.


## 9. Ordering at Glass-Liquid Transition

Because
of the formation of percolation cluster at glass-liquid transition, the amorphous
material looses the invariance for the Euclidian space isometries such as
translation and rotation on length scales smaller thanξ(T), for example, at these length scales, it is
dynamically inhomogeneous. The percolation cluster is also called an infinite
cluster as it penetrates the whole volume of material which as a result
drastically changes its physical properties from solid-like below to fluid-like
above the percolation threshold. The geometry of a percolation cluster near the
percolation threshold is fractal with the Hausdorff-Besikovitch dimension df=d−β/ν,
where β and ν are critical exponents (indexes) and d = 3 is
the dimension of the space occupied by the initial-disordered network, so that df=2.55±0.05.
The formation of percolation cluster changes the topology of bonds network from
the 3D Euclidean below to the fractal df-dimensional above the percolation threshold. At
glass-liquid transition, the amorphous material changes the group of isometries
from the Euclidian to the fractal space group of isometries at length scales
smaller than ξ(T). An
amorphous material is represented by a disordered bond network at all
temperatures, however it has a uniform 3D distribution of network breaking
defects at low concentrations in the glassy state and a fractal df-dimensional distribution at high enough
temperatures when their concentration exceeds the percolation threshold in the
liquid state. Although on average the liquids are homogeneous, they are
dynamically inhomogeneous on length scales smaller than ξ(T) near the glass transition. Changes that
occur in the geometries of amorphous material at Tg affect their
mechanical properties. Above Tg, the geometry is fractal like in
liquids [73] and the mechanical properties are similar to those of liquids. The
structure of material remains disordered at all temperatures although the space
distribution of configurons as seen above is different below and above the
percolation threshold changing the geometry from the Euclidean to fractal.Although,
to a certain extent, being disordered at all temperatures, the bond structure
above the percolation threshold becomes more ordered as a significant fraction
of broken bonds, for example, configurons belong to the percolation cluster. The density of the percolation cluster of configurons,φ, can serve as
the order parameter [6, 99] and for the liquid phase it has nonzero values,
whereas for the glassy phase, φ = 0 (Figure 14).Temperature dependence
of the order parameter of configurons in (a) amorphous SiO2 (b) and GeO2 [6]. The liquid phase
seems more ordered for configurons (broken bonds) which are largely a part of
the percolation cluster in the liquid whereas in the glassy phase the
configurons are randomly distributed in the solid.
(a)(b)Second-order phase transitions in ordered substances are typically
associated with a change in the crystal lattice symmetry, and the symmetry is
lower in the ordered phase than in the less ordered phase [98]. In the spirit of the Landau ideas, the transition from a glass to a liquid
spontaneously breaks the symmetry of bonds, for example, of the configuron
system. At glass-liquid transition, the amorphous
material changes the group of isometries from the Euclidian to the fractal
space group of isometries at length scales smaller than ξ(T). The description of a
second-order phase transition as a consequence of a change in symmetry is given
by the Landau-Ginzburg theory [100]. The order parameter φ, which equals
zero in the disordered phase and has a finite value in the ordered phase, plays
an important role in the theory of second-order phase transitions. For a glass-liquid
transition, as well as in the general case of percolation phase transitions, the density of the percolation cluster of configurons is an order parameter [6].Crystalline
materials are characterised by 3D Euclidean geometries below their melting
pointTm. Thus, both glasses below Tg and crystals below
Tm are characterised by the same 3D geometry. Glasses behave like
isotropic solids and are brittle. Because of the 3D bond geometry, glasses break
abruptly and the fracture surfaces of glasses typically appear flat in the
“mirror”  zone. Glasses change their bond geometry at Tg. When melting occurs, the geometry of crystalline materials also changes,
as revealed by MD experiments to fractal structure with df≈2.6 [73]. It is
also known that emulsion particles have homogeneous fractal distribution in
liquids and the fractal dimension of emulsions is df=2.5±0.1  [101]. With
the increase of temperature, the clusters of configurons grow in size whereas
clusters of atoms decrease their sizes. Further changes in the dimensionality
of bond structure can occur. Finally, when no unbroken bonds remain in the
system, the material is transformed to a gaseous state. Therefore, for the
system of bonds the phase changes can be represented by the consequence of
changes of the Hausdorf dimension d = 3 (solid)→df = 2.55±0.05  (liquid) →d =  0 (gas).


## 10. Conclusions

Amorphous
materials can occur either in liquid or glassy state. Amorphous materials are largely
spread in the nature both in form of melts and glasses, for example, it has
been found that the inner core of Earth is in a disordered, for example, glassy
form [102] (see also [103]). The transition from the liquid to the glassy state
evidences characteristic discontinuities of derivative
thermodynamic parameters such as the coefficient of
thermal expansion or the specific heat. The
analysis of bonding system of glassy and crystalline materials shows that they both
hold the same Hausdorff dimension of bonds d = 3. The similarity in bonding of
both glassy and crystalline materials means the similarity of their behaviour
in many aspects such as the propagation of acoustic signals which revealed the
solid state of the Earth core.Amorphous
materials are liquid above the glass-transition temperature. The configuron
model of glass transition shows that the transition of amorphous materials from
glassy to liquid state is a percolation-type phase transition. The bonding
system of an amorphous material changes its geometry from 3D in the glassy
state to fractal one (df=2.55±0.05) in the liquid state due to formation of infinite size dynamic percolation
clusters made of broken bonds—configurons. The
configuron model of glass transition gives an explicit equation of glass-transition
temperature (29) and demonstrates characteristic jumps in specific heat and
thermal expansion. The higher the concentration of broken bonds is, the lower
the viscosity is, which is a continuous function of temperature both for glassy
and liquid amorphous materials and has no discontinuities at glass transition.
The defect model of viscosity results in a universal viscosity (12) valid at
all temperatures. Table 7 summarises temperature changes of states, geometry of
bonds, and viscosity of amorphous materials.Table 7
Viscosity and bond geometries of amorphous
materials.
TemperatureT<TgT>TgStateSolid (glassy)Liquid (melt)The
Hausdorffd = 3df=2.55±0.05dimension ofbondsViscosityContinuous decreasing function of temperatureActivation energy highActivation energy lowTransitions
in disordered media from glassy to liquid states are universal and reflect
changes in the bonding system. Because of that, the configuron model of glass
transition can be used to provide insights on embrittlement of materials
composed of microcrystals at low temperatures as well as on such natural
phenomena as quick sand formation. In all such cases, formation of additional
bonds between elementary particles which constitute the material, for example, microcrystals
or sand grains leads to their solid-like behaviour at lower temperatures or
denser packing.



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*Source: 817829-2009-03-15.xml*