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Advances in Condensed Matter Physics
Volume 2008 (2008), Article ID 817829, 23 pages
http://dx.doi.org/10.1155/2008/817829
Review Article

Viscosity and Glass Transition in Amorphous Oxides

Department of Engineering Materials, The University of Sheffield, Sir Robert Hadfield Building, Mappin Street, Sheffield S1 3JD, UK

Received 12 June 2008; Revised 20 October 2008; Accepted 7 December 2008

Academic Editor: Mark Bowick

Copyright © 2008 Michael I. Ojovan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [13], 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 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].

tab1
Table 1: Commercial oxide glass compositions.

3. Melting of Amorphous Solids

Crystalline materials melt at well-defined melting temperatures whereas amorphous materials transform from glassy, for example, solid form to liquid-state at glass transition temperatures 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 [2530]. A glass is most commonly formed by cooling a viscous liquid fast enough to avoid crystallisation. Compared with , the actual values of 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).

fig1
Figure 1: Specific heat of amorphous o-terphenil. (a) Experimental and (b) calculated [12].

These discontinuities allow detecting the [31] or, accounting for cooling rate dependences, the glass transition interval where a supercooled liquid transforms to a glass (Figure 2).

817829.fig.002
Figure 2: Determination of glass transition temperature based on differential temperature analysis (DTA), below the the material is glassy whereas above the 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 depending on the cooling rate with maximal 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 , whereas the systems are completely frozen with respect to primary relaxation at [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)).

fig3
Figure 3: (a) Regions of relaxation behaviour of amorphous materials (after [29]); (b) density and geometry of bonds on increase of temperature (after [6]).

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 Pas (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: 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 subset A 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 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 Roughly, if N(λ) grows proportionally to 1/ 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 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) -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.

tab2
Table 2: Viscosity of some amorphous materials.

It is commonly assumed that shear viscosity is a thermally activated process. Since the pioneering work of Frenkel [44] fluid viscosity, , has been expressed in terms of an activation energy Q by 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, or , 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: 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  Pa [45], hence at η = 1012 Pas, where the practical purpose (relaxation) glass transition occurs (see (1)), the Maxwell relaxation time seconds. Accounting that for fused silica the activation energy of viscosity at low temperatures = 759 kJ/mol and the shear modulus of fused silica is 31 GPa at 25 [46, 47], one can see that the relaxation time at STP becomes as long as 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 (Figure 4).

817829.fig.004
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 where are additivity factors for a given component , and 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, typically when viscosity is 1.5, 6.6, and 12 (point of practical purpose glass transition) [5153]. A detailed overview on statistical analysis of viscosities and individual oxide coefficients 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, 5154]. 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 (isokom) [53].

fig5
Figure 5: Effect of component addition on isokom log[] = 1.5 [53]. Courtesy Alexander Fluegel.

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 Pas. This definition of was used by Angell to plot the logarithms of viscosity as a function of () [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 whereas at high temperatures the activation energy is low . 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]: The Doremus fragility ratio ranges from 1.45 for silica to 4.52 for anorthite melts (Table 3).

tab3
Table 3: Asymptotic Arrhenian activation energies for viscosity and the corresponding Doremus fragility ratios [47].

The higher the value of 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 , and fragile materials if they have . 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)).

817829.fig.006
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 Figure 6) over many orders of magnitude with a high accuracy: where , , and (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 molecule . 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]: where is the volume of a molecule, and B are constants. In terms of the specific volume V per molecule, it can be shown that , for some constant and low temperatures . Clearly, (8) is the same as (7) when ,   and . 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]: where and are adjustable constants and is the configurational entropy. Assuming that , where is the relaxational part of the specific heat, one can see that (9) transforms into VTF (7), where and . 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 Figure 6). 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]: where in this case is defined by , 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 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 and . For the low viscosity range (), Volf gives = 80–300 kJ/mol and for the high viscosity range and = 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 , where designates a bond between Si and O, and—designates a bridging oxygen atom with two bonds . 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 where    is the formation Gibbs-free energy, is the enthalpy, is the entropy, and 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 , 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 , where is the Gibbs-free energy of motion of a jumping configuron, and 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] with where , f is the correlation factor, g is a geometrical factor (approx. 1/6), λ is the average jump length, 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 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 (usually the case) and letting . 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 , the viscosity of melts changes to a nonactivated, for example, non-Arrhenius behaviour, which is characteristic of systems of almost free particles.

The five coefficients , , 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).

fig7
Figure 7: (a) Viscosity-temperature data for anorthite and (b) for diopside, where calculated curves were obtained using (12).

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 , , 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.

tab4
Table 4: Thermodynamic parameters of configurons.

Data from Table 4 show that for most materials examined the entropy of configuron formation is significantly higher than the entropy of configuron motion . Taking into account the values of this means that 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, 6668]. 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 [13, 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, 7173]. 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 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 [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.

tab5
Table 5: Basic glass transition models.
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 above in an amount given by where is the molar volume at , is the change in the volume expansion coefficient which occur at [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 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 are untenable for free volume model as the free volume contraction could not explain production of relative mobile liquids above .

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 length ξ of molecular motion increases with the decrease of temperature [61]. The structural relaxation time depends on the configurational entropy as where and C are constants. It is then assumed that , where is the relaxational part of the specific heat, 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 , which for typical glass formers is . For , 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 . 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).

817829.fig.008
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 is ξ. 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 temperature 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 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, , 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 where   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, , which results from the symmetry-selective parts of the interactions and increases the quality of bonds. The average fraction of LFS () is given by where is the Boltzmann constant, P is the pressure, and are the energy gain and the volume change upon the formation of an LFS, and are the degrees of degeneracy of the states of LFS and NLS, respectively. It is assumed that and . can be either positive or negative depending upon material, for example, 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 NLS is a short-lived random structure whereas LFS a long-lived rigid structural element. The lifetime of an LFS can be estimated as where is the structural relaxation time of NLS and 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 the 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 (), it increases the fragility of SiO2 and weaken the boson peak [80].

817829.fig.009
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 [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 [8386]. 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] where and 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.

fig10
Figure 10: 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 . (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.
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 or above a corresponding packing fraction, so that above the dynamics is liquid-like whereas below 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 to the experimental time taken arbitrary as = 100 seconds [88]. The Hunt equation relates the glass transition temperature with the peak energy in the distribution of hoping barrier heights of individual relaxation processes where 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 : where and are ionic charges, 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

where P is pressure, Hunt obtained an excellent description of pressure dependence of the glass transition temperature in ionic liquids [76, 89]. Moreover, this approach gave an explanation of reduced glass transition temperature by confinement in small pores [90]: 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 , 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, . Figures 11(a) and 11(b) illustrate schematically the replacement of a disordered atomic structure by the congruent bond structure.

fig11
Figure 11: Schematic of disordered bond lattice model of an amorphous material: (a) distribution of atoms in amorphous phase at ; (b) distribution of bonds in amorphous phase at ; (c) distribution of bonds in amorphous phase at ; (d) distribution of bonds in amorphous phase at higher temperatures when configuron clustering occurs.

At absolute zero temperature , 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 . 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: 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.

fig12
Figure 12: (a) Schematic of 4 covalent bonds (b) and one broken bond in SiO2.

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 interval governing (27) [64]. The statistics of two level systems leads to the well-known relationship for equilibrium concentrations of configurons and unbroken bonds [64, 94] at absolute zero temperature . The concentration of configurons gradually increases with the increase of temperature, and at   achieves its maximum possible value if . At temperatures close to absolute zero, the concentration of configurons is very small . 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 the (where the amorphous material is solid), to fractal above the , 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 density , which remains the same for both ordered and disordered lattices [95, 96]. The relative concentration of broken bonds is given by which shows that the higher the temperature is, the higher is . Assuming that at the space is completely filled by configurons, one can designate as the volume fraction of space occupied by configurons. Thus, the critical (glass-transition) temperature at which the percolation threshold is achieved can be found assuming that the configurons achieve the universal critical density given by the percolation theory For SiO2 and GeO2, it was supposed that , where is the Scher-Zallen critical density in the 3D space [9597]. For many percolating systems, the value of is significantly lower compared to [96].

At temperatures above , the space is filled by configurons at concentrations which exceed the critical density , 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]: Note that because , this equation can be simplified to , which is an analogue of the Diennes ratio used to assess the melting temperatures of crystalline solids. Below the , 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 , the concentration of defects achieves the critical concentration for formation of a percolation cluster. Above the , a percolation cluster made of configurons is formed, and the geometry of the network becomes fractal with the fractal dimension near the .

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] , where is the valence of the cation-anion pair, is the standard unit of charge, 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 length . Below the , the correlation length gives characteristic sizes of clusters made of configurons whereas above the , it shows characteristic sizes of clusters made of unbroken bonds, for example, atoms. gives the linear dimension above which the material is homogeneous, for example, a material with sizes larger than has on average uniformly distributed configurons. At sizes smaller compared to , 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 . 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 . The liquid near the glass transition is dynamically inhomogeneous on length scales smaller than and remains unchanged for fractal space group of isometries. Figure 13 shows schematically the structure of an amorphous material near the glass transition temperature.

817829.fig.013
Figure 13: Schematic representation of the dynamic homogeneous fractal structure of an amorphous material near the glass transition temperature. Below the , the correlation length gives characteristic sizes of clusters made of configurons, whereas above the , 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 is . At temperatures far from , the correlation length is small, whereas at temperatures approaching , it diverges where the critical exponent ν = 0.88 [93, 94].

If sample sizes are smaller than , 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 ; when sample sizes L  decrease [6], 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 was found as [6, 12] where , β = 0.41 is the critical exponent [95, 96], is a numerical coefficient close to unity, for example, for strong liquids = 1.0695, and is the enthalpy of configurons in the percolation cluster.

The configuron model of glass transition shows that the linear expansion coefficient near the behaves as [6] where 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 proportional to [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 of in these systems is significantly lower compared to strong materials as can be seen from Table 6 [47].

tab6
Table 6: Glass transition temperatures of amorphous materials.

Data from Table 6 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 to 3, occurs in fragile liquids at lower percolation threshold compared to strong liquids. The effective configuron radius, , can be assessed from where is the bond radius (half of bond length). For strong liquids, 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, . For example, B2O3 with fragility ratio = 3.28 has , 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 than , 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 , 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 . The formation of percolation cluster changes the topology of bonds network from the 3D Euclidean below to the fractal -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 . 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 -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 near the glass transition. Changes that occur in the geometries of amorphous material at affect their mechanical properties. Above , 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, , can serve as the order parameter [6, 99] and for the liquid phase it has nonzero values, whereas for the glassy phase, = 0 (Figure 14).

fig14
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.

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 . 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 , 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 point . Thus, both glasses below and crystals below 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 . When melting occurs, the geometry of crystalline materials also changes, as revealed by MD experiments to fractal structure with [73]. It is also known that emulsion particles have homogeneous fractal distribution in liquids and the fractal dimension of emulsions is [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) = (liquid) = 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 () 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.

tab7
Table 7: Viscosity and bond geometries of amorphous materials.

Transitions 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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