On relaxation nature of glass transition in amorphous materials

Damba S. Sanditov1,2 and Michael I. Ojovan3,4[footnoteRef:2] [2: Department of Materials, Imperial College London, SW7 2AZ, United Kingdom, e-mail: m.ojovan@imperial.ac.uk; ]

1Buryat State University, Ulan-Ude, 670000, Russia, 2Institute of Physical Material Science, Siberian Branch, Russian Academy of Sciences, Ulan-Ude, 670047. Russia, e-mail: sanditov@bsu.ru

3Department of Materials, Imperial College London, SW7 2AZ, United Kingdom, e-mail: m.ojovan@imperial.ac.uk; 4Department of Radiochemistry, Lomonosov Moscow State University, Moscow, 119991, Russia, e-mail: m.i.ojovan@radio.chem.msu.ru

Abstract

A short review on relaxation theories of glass transition is presented. The main attention is paid to modern aspects of the glass transition equation qτg = C, suggested by Bartenev in 1951 (q – cooling rate of the melt, gstructural relaxation time at the glass transition temperature Tg). This equation represents a criterion of structural relaxation at transition from liquid to glass at T = Tg (analogous to the condition of mechanical relaxation ωτ = 1, where the maximum of mechanical loss is observed). The empirical parameter С = Tg has the meaning of temperature range Tg that characterizes the liquid-glass transition. Different approaches of Tg calculation are reviewed. In the framework of the model of delocalized atoms a modified kinetic criterion of glass transition is proposed (q/Tg)g= Cg, where Cg 7·10-3 is a practically universal dimensionless constant. It depends on fraction of fluctuation volume fg, which is frozen at the glass transition temperature Cg = fg/ln(1/fg). The value of fg is approximately constant fg 0.025. At Tg the process of atom delocalization, i.e. its displacement from the equilibrium position, is frozen. In silicate glasses atom delocalization is reduced to critical displacement of bridge oxygen atom in Si-O-Si bridge necessary to switch a valence bond according to Muller and Nemilov.

An equation is derived for the temperature dependence of viscosity of glass-forming liquids in the wide temperature range, including the liquid-glass transition and the region of higher temperatures. Notion of (bridge) atom delocalization is developed, which is related to necessity of local low activation deformation of structural network for realization of elementary act of viscous flow – activated switch of a valence (bridge) bond. Without atom delocalization (“trigger mechanism”) a switch of the valence bond is impossible and, consequently, the viscous flow. Thus the freezing of atom delocalization process at low temperatures, around Tg, leads to the cease of the viscous flow and transition of a melt to a glassy state. This occurs when the energy of disordered lattice thermal vibrations averaged to one atom becomes equal or less than the energy of atom delocalization.

The Bartenev equation for cooling rate dependence of glass transition temperature Tg = Tg(q) is discussed. The value of fg calculated from the data on the Tg(q) dependence coincides with result of the fg calculation using the data on viscosity near the glass transition. Derivation of the Bartenev equation with the account of temperature dependence of activation energy of glass transition process is considered. The obtained generalized relation describes the Tg(q) dependence in a wider interval of the cooling rate compared Bartenev equation. Experimental data related to standard cooling rate q = 3 K/min were used in this work.

Keywords: glass transition, relaxation time, cooling rate, delocalized atoms, configurons, viscous flow

~ 1 ~

2

3

1. Background

Glasses are solid amorphous materials which transform into liquids upon heating through the glass transition e.g. the solid-like behaviour of glasses is separated from liquid-like behaviour at higher temperatures by the glass transition temperature, Tg. The glass transition has a pronounced relaxation, kinetic character [1-7] although it is similar to a second-order phase transition in the Ehrenfest sense with continuity of volume and entropy, and discontinuity of their derivatives which are used in practice to detect Tg [8-12]). Discussion of nature of glass continues [13-16] and after some lull it increases significantly, especially in the second decade of the new century as the microscopic mechanism generating the glassy state of matter is still debated. Although specified in the UIPAC definition of glass transition as a second order phase transformation [17] glasses are most often considered as just extremely viscous liquids rather than resulting from any kind of thermodynamic phase transition thus experimental evidences of phase transition at Tg, such as the specific heat jump, large change of the thermal expansion coefficient and the absence of variation of Tg at low cooling rates are mostly ignored. Decent models treating microscopic mechanisms behind slowing down of relaxation at Tg and describing transitions in liquids and glass transition as true phase transformation have been nevertheless developed among which one can see for example [18 - 26]. Tournier e.g. considers the transition at Tg as due to a change of the undercooled-liquid Gibbs free energy, which is the driving force of the glass transition. The classical Gibbs free energy change for a crystal formation has been completed with an enthalpy saving which allowed a description of liquid-liquid and stable glass transitions. Most important is that both percolation-based models [22, 24-26] and the Tournier model of glass transition [18-21, 23] correctly predicts the specific heat jump, the large change of the thermal expansion coefficient at Tg. Moreover newer experimental evidences have been recently found revealing thermodynamic (although kinetically controlled) nature of glass transition such as direct visualisation of macroscopic percolating clusters formed by molecules at glass transition as predicted by theoretical models [27], clustering [28] and structural changes upon glass-transition revealed by in-situ studies of glass-transition by synchrotron XRD via reciprocal and real-space radial distribution functions [29, 30], and high-precision measurements of third- and fifth-order nonlinear dielectric susceptibilities that strongly support theories based on thermodynamic amorphous order which is fractal by dimension [31]. The glass transition may therefore belong to a currently developed class of critical phenomena generically termed topological phase transitions which are amenable to the scaling approach and characterised by diverging length and time at the transition [32] operating however with fractional dimensions of the conventional space of joining bonds of matter [22, 24, 25, 27, 33, 34].

The nature of glass transition remains nevertheless one of the topical non-solved problems of the condensed matter physics. Unlike crystal or liquid glassy solid state is in non-equilibrium state, which happens to be (meta)stable, since the transition into equilibrium state is restricted by activation barrier. Both ancient inorganic (silicate) glasses found in natural environments fossil and amber can be given as examples that kept the amorphous structure for tens of millions of years. During external actions, e.g. during annealing, a slow, but continuous density increase is observed – its volume relaxation, which reflects the tendency of the system to the equilibrium state. This paper is devoted to discussion of the modern aspects of relaxation theory of glass transition and viscous flow of glass-forming liquids. New results using this model were obtained, and development of this model is considered.

1. Bartenev’s approach 1.1. The empirical equation of glass transition

The structure of matter freezes during the glass transition, corresponding to conditions, at which molecular rearrangements become so slow (relaxation time becomes so long), that structural changes cannot follow changes of external parameters (in this case temperature). Thus it follows naturally that the glass transition temperature Tg depends on the cooling rate of the glass-forming melt q = dT/dt. The lower the cooling rate, the lower the glass transition temperature (Figure 1.1).

Figure 1.1.Volume change during the liquid –glass transition in the process of cooling. ΔTg – glass transition region, Tg – glass transition temperature. Tg1 – corresponds to cooling rate q1, and Tg2 – to cooling rateq2, q2

From these and other experimental data it follows that glass transition is a relaxation process and obeys kinetic laws. Evolution of a glass-forming system depends on the rate of changes of external parameters (temperature, pressure), as well as on the relaxation time of the system to the corresponding equilibrium state. Thus, from the point of relaxation approach in the process of liquid-glass transition a deciding role is played by the relation between relaxation time and cooling rate q [5-7, 35-37]. In 1951 Bartenev [37] on the basis of general considerations has suggested the following kinetic criterion of glass transition

,(1.1)

where g is the relaxation time at the glass transition temperature Tg, and С is an empirical parameter with the dimension of temperature.

Relation (1.1), which is sometimes called main equation of glass transition [38-40], is successfully applied in relaxation spectrometry of polymers and glasses [39, 40] as a condition of structural relaxation transition at T = Tg and is an analogous of usage of criterion w = 1 during mechanical relaxation, at which the maximum of mechanical losses is observed. Equations like (1.1) are as well used for description of other relaxation processes, e.g. for thermosimulated electric depolarization of amorphous polymers [39] (e.g. qi = Ci, where i is the relaxation time of i-th relaxation process).

The transition of a liquid into a glassy state upon cooling (and also under the action of high pressure) is called structural glass transition, and the transition from a viscous Newtonian fluid to an elastic glassy body under periodic mechanical action with a certain frequency ν is called dynamic (sometimes mechanical) glass transition [39]. The Bartenev equation (1.1) is a condition for the realization of structural glass transition, which is analogous to the criterion of dynamic glass transition at temperature Tν [39, 41]

.(1.2)

By excluding relaxation time g from relations (1.1) and (1.2), it is possible to obtain the frequency ν=νequiv, which is equivalent to a given cooling rate q [39]

.(1.3)

At С = qg this equation becomes a more convenient known formula [41]

. (1.4)

The structural relaxation time g at Tg of inorganic glasses is of the order of [38-40, 42]

, (1.5)

Accounting for that, we conclude that in accordance with equation (1.4) the equivalent frequency of inorganic glasses is

Hz.(1.6)

This result means that only at low frequencies, of the order of 10-3 Hz, the temperature of the dynamic vitrification Tν coincides with the temperature of the structural glass transition Tg. Both structural and dynamic vitrification occur simultaneously: the structural (topological) and viscous components of the deformation are frozen at the same time [39].

1.2. Dependence of the glass transition temperature on the cooling rate of the melt

The dependence of Tg on melt cooling rate is an important part of the theory of glassy state. It is also important for practical applications, for example, when developing optimal modes for annealing of glasses. By substituting into the glass transition equation (1.1) the relaxation time g from well-known formula (Frenkel equation)

,(1.7)

written at T=Tg and τ=τg, Bartenev [37] has obtained the following dependence of Tg on the cooling rate

,(1.8)

, (1.9)

, (1.10)

where a1 and a2 are empirical constants, τ0 is the vibration period of a molecule, U is the activation energy of vitrification process, R is the gas constant, and С is the empirical parameter of glass transition of equation (1.1). The value of U was assumed as constant independent of temperature, e.g. U = const, while deriving (1.8).

In 1954, three years later, Ritland proposed a relation similar to (1.8) using other than Bartenev initial premises [43]. Therefore, the equality (1.8) is sometimes called the Bartenev-Rittland equation [1, 2].

The Tg is measured both in the heating mode during glass softening and in the cooling mode of glass-forming melts. It turns out that although the values of Tg obtained under these regimes are somewhat different, the dependence of the Tg on heating rate is approximately the same as the dependence of Tg on cooling rate. Bartenev [37] carried out a verification of dependence (1.8) with heating along the curve of thermal expansion of silicate glasses, and also on cooling along the quench curve of the same glasses. The experimental points in both regimes lie on the straight line in the coordinates (1/Tg) – logq corresponding to equation (1.8). The linear dependence of 1/Tg on logq is observed for a wide variety of glasses, (see for example figure 1.2) [38, 44-47], including metallic glasses [48].

Figure 1.2. Dependence of the reciprocal value of Tg in the logarithm of cooling rate log q of rubbers SKS-30 (1), SKN-18 (2) and SKN-40 (3) [44].

At the same time, a deviation from the Bartenev equation (1.8) was found for a number of glasses at relatively high heating (cooling) rates (see figure 1.3) [45].

Figure 1.3. Dependence of Tg on cooling rate of melt for lead-silicate glasses №1 and №2 in coordinates 1/Tg – log q according to data from Bartenev and Lukyanov [45].

Bartenev and co-workers [44, 45] carried out a systematic study of dependences Tg= Tg(q) over a wide range of heating rates within 0.2 – 50 K/min. Various amorphous substances have been studied: rosin, ebonite, organic amorphous polymers, silicate glasses with Tg from 208 to 1025 K (Table 1.1).

Table 1.1. Parameters of Bartenev equation (1.8) and fraction of fluctuation volume fg at Tg [44, 45].

Glass

Tg, K

a1103, K-1

a2105, K-1

fg, see equation (10.4)

Rosin

313

3.098

8.3

0.027

40

0.025

PS

345

2.78

9.0

0.032

34

0.029

PMMA

349

2.75

8.9

0.032

34

0.029

RubberSKS-30

208

4.62

15.0

0.032

34

0.029

Rubber SKN-18

218

4.41

13.2

0.030

36

0.028

Rubber SKN-40

246

3.90

12.0

0.031

35

0.028

Ebonite

349

2.72

9.6

0.035

31

0.032

Boric anhydride

534

1.81

5.6

0.031

35

0.028

Silicate glass* (mass %)

№ 1

714

1.34

4.28

0.032

34

0.029

№ 2

744

1.29

4.24

0.033

33

0.030

№ 3

809

1.19

3.60

0.030

36

0.028

№ 4

885

1.086

3.33

0.031

35

0.028

№ 5

1025

0.94

2.67

0.028

39

0.026

* Silicateglasses: № 1: SiO2 – 55.3, Na2O – 3.8, K2O – 9.2, PbO – 30, Al2O3 – 1.7; № 2: SiO2 – 38.1, Na2O – 1.3, K2O – 2.5, PbO – 52, Al2O3 – 3.4, B2O3 – 1.8, CaO – 0.5, MgO – 0.4; № 3: SiO2 – 70.9, Na2O – 16.1, K2O – 0.6, CaO – 8.1, MgO – 2.9, other oxides – 1,4; № 4: SiO2 – 56, Na2O – 10.1, CaO – 17, MgO – 4, Al2O3 – 11, B2O3 – 2; № 5: SiO2 – 57.6, CaO – 7.4, MgO – 8, K2O – 2, Al2O3 – 25. PS – polystyrene, PMMA – polymethylmethacrylate. Values а1 and а2 correspond to the case when logarithm in Bartenev equation (1.8) is decimal (logq).

The Bartenev equation (1.8) has been shown valid with rare exceptions. It has been also established [38, 44, 45] that the ratio of parameters a1 and a2 is practically constant (a2/a1 ≈ 0.03) for various amorphous substances (see Table 1.1 and Figure 1.4)

.(1.11)

Figure 1.4. Linear correlation between Bartenev equation parameters a1and a2. 1 – inorganic silicate glasses; 2 – amorphous organic polymers.

Table 1.2 shows the results of our study of the oxygen-free glasses Se – Ga, Se – Bi, In – Se, As – Sb – Se. Data from the SciGlass database were also used [49].

Table 1.2. Bartenev equation parameters (1.8) and fraction of fluctuation volume fg for anoxic and a number of oxide glasses [49]

Glass

Tg, K

a1 · 103, K-1

a2 · 105, K-1

a2/a1

fg

Se - Bi

309

3.12

10.45

0.034

0.031

Se - Ga

315

3.09

9.29

0.030

0.027

In-Se

316.7

3.09

5.54

0.018

0.017

As - Sb - Se

444.8

2.15

6.74

0.031

0.028

GeO2

762

1.23

3.45

0.028

0.026

P2O5- TeO2

578

1.67

4.99

0.030

0.027

SiO2 - Al2O3 - B2O3 - P2O5 - MgO - Na2O - K2O

1064

0.89

3.20

0.036

0.032

The dependence of 1/Tg on lnq turns out to be linear (Figure 1.5), which confirms the applicability of equation (1.8) to these and other systems (Figure 1.6). For oxygen-free glasses, the same regularities are observed as for oxide glasses: a2/a1 ≈ 0.028–0.034. Only the In-Se glass falls out of the general pattern.

Figure 1.5. Dependence of the reciprocal glass transition temperature on heating rate for anoxic glasses in coordinates 1/Tg – lnq. Se/Bi, mol.%: 77.28/22.72; Se/Ga, mol.%: 95.56/4.44; In/Se, mol.%: 7.11/92.89; As/Sb/Se, mol.%: 32.91/7.64/59.45. Data from [49]: Se - Bi [(25097) Abu El-Oyoun M., 2000], Se - Ga (26436) El-Oyoun M.A., 2003], In - Se [(26416) Abd El-Moiz A.B, 1992], As – Sb - Se [(25427) Mahadevan S., 1986].

Figure 1.6. Dependence of the reciprocal glass transition temperature on heating rate for oxide glasses in coordinates 1/Tg – lnq. SiO2/Al2O3/B2O3/P2O5/MgO/Na2O/K2O, mol.%: 55,09/22,01/1,01/ 1,72/19,78/0,32/0,0507; GeO2, mol.%: 100; P2O5/TeO2, mol.%: 69,39/30,61. Data from [49]: SiO2 - Al2O3 - B2O3 - P2O5 - MgO - Na2O - K2O [(24241) Watanabe K., 1994], GeO2 [(14972) Bruning R., 1999], P2O5 - TeO2 [(8772) Elkholy M.M., 1995].

In the work of Bartenev and Lukyanov [45], the dependence of Tg of amorphous substances on heating rate was studied by the method of thermal linear expansion. Samples in the form of round rods 4 mm in diameter and 50 mm long were placed in the middle part of a tubular electric furnace having a length of 500 mm. The temperature along the sample was the same with an accuracy of one degree. Errors in the measurement of temperature were 0.5 degrees for polymers and 1 degree for silicate glasses. The accuracy of measuring the elongation of the sample during expansion was 1-2 micrometers. To exclude the influence of thermal prehistory, carefully annealed samples were used. Bartenev and Gorbatkin [44] studied the dependence of Tg of rubbers on the rate of cooling (see Figure 1.2). The length of the samples was measured with an accuracy of 1.5–2 micrometers. The cooling rate varied from 0.3 to 30 K/min.

2. Mandelstamm – Leontovich theory

In addition to parameters that characterize the state of a system in classical thermodynamical equilibrium new internal (structural) order parameters are introduced to describe nonequilibrium state in the framework of approach based on nonequilibrium thermodynamics (de Donde [50], Prigogine, Defay [51]). Consideration of transition of a liquid into a glass is from these positions a development of the ideas formulated by Bragg and Williams [52] in 1934 and Mandelstam and Leontovich [41] in 1937. To describe the glass transition of a liquid or, conversely, transition of a glass into a liquid, at least one additional internal parameter ξ is introduced, which describes the transition of system to a nonequilibrium state [41]. If the state of the system in equilibrium is completely determined by two independent variables Р and Т, for the equilibrium value of the parameter ξ=ξe we have

.

However, if Р and Т change, then the new equilibrium value of ξ is not reached instantaneously, but at some finite rate, so in the general case ξ≠ ξe. In such a nonequilibrium process, the parameter ξ is a new independent variable that determines the state of the system. For example, a change of ξ can lead to a change in the volume of the system. Then the equation of state is written in the form

.(2.1)

If we determine the parameter ξ using the "internal" increment of entropy, corresponding to an irreversible process of establishing equilibrium,

,

then, for example, the differential of the Gibbs thermodynamic potential takes the following form:

,.

Further, for the thermodynamic system of equations to be closed, it is necessary to introduce a relaxation equation for the quantity ξ. For small deviations from equilibrium, it has the following form [41]

,(2.2)

where is relaxation time of the system.

The problem of describing relaxation processes therefore consists of a proper choice of parameter ξ, which is able to characterize the nonequilibrium structure of matter in the glass transition region, and in the formulation of equations describing its relaxation. The well-known theories are those where fraction of particles n2 in the excited state (the theory of Volkenstein-Ptitsyn [6, 7] and Gotlib-Ptitsyn [53]), fictive temperature Т* [54], and free volume fraction ƒ [55] serve as an internal parameter ξ.

Mandelshtam and Leontovich [41] considered a periodic (sinusoidal) mechanical action on the equilibrium liquid, which is created by a sound wave with a circular frequency ω. This effect creates regions of local contraction and rarefaction, characterized by deviations in the density (∆ρ) from the system average, temperature deviations (∆Т) and structural (internal) parameter (∆ξ) from their equilibrium values. The Mandelstam-Leontovich theory has a rigorous physical basis and is mathematically clearly formulated in terms of the response of system's properties to external action. In addition, this theory serves as a starting point in the construction of a system of equations for nonequilibrium thermodynamics (see the review [56]).

On the basis of the Mandelstam-Leontovich theory, following Nemilov [14], we consider establishing a relationship between the melt cooling rate q and the structure relaxation time g at Tg [i.e. in the form of an expression analogous to the Bartenev equation (1.1)]. The relation between the theory describing the appearance of a solid-like state in a liquid can be written in the form [41]

.(2.3)

Here P∞ is value of a certain property corresponding to solid-like behavior of liquid, ω is the circular frequency of the external action (in the general case, the operator determining the temporal regime of the effect on the system), ∆P is the contribution to the property introduced by the structural changes having a relaxation character (relaxation contribution).

Since the solid-like behavior of an amorphous material in modern representations corresponds to a glassy state, this theory is also applicable to the description of this state. The condition for the liquid-glass transition in this version of the kinetic theory of vitrification [56] is the equality (1.2)

. (2.4)

In accordance with (2.3), the transition from the equilibrium liquid to the glass is realized not at the point Tg, but within a certain temperature interval. Suppose that there is a periodic sinusoidal temperature change (Figure 2.1)

.(2.5)

Figure 2.1. To the explanation of the interrelation between periodic temperature change and its approximation in the form of linear function on certain intervals. 1 – sinusoid, equation (2.5); 2 – linear parts of the curve as corresponding to approximation dT/dt = ΔT0·ω·cos(ωt) (see text).

The part of the sinusoid on a sufficiently wide time interval t can be considered as a linear dependence of Т with time t. On this segment, the temperature change is approximately ∆То [35]. The rate of temperature change is found by differentiating the relation (2.5) with ω = const

.

Then for ωt multiples to π (within the indicated variations of ωt), the cosine value is equal to |cos ωt| ≡ 1, whence for q = dT/dt we have

.(2.6)

This equality coincides with (1.3) for С = ∆To, where С is the empirical parameter of the Bartenev equation (1.1) with the temperature dimension. Substituting ω from this expression into the glass transition condition (2.4), we obtain the relationship between the cooling rate of the melt q and the structural relaxation time τg at the glass transition temperature Tg [35],

,(2.7)

which can be regarded as a justification of the glass transition equation (1.1).

Here, ∆To is a certain scaling factor, which allows to approximately consider the linear temperature change in the range ∆To as a part of an extended periodic process with a frequency ω. Since outside the glass transition interval ∆Tg the external effect on the system with frequency ω does not leave any changes in the system, ∆To has the meaning of the temperature range in which glass transition actually occurs.

In fact, at high temperatures of system, when the relaxation time is short and the periodic changes ∆To do not follow the condition (2.6), the liquid is in a metastable state and all local changes in Т do not appear in its properties, since the state does not freeze, all the regions of the system return to the equilibrium state: condition ω< 1. The time of structure relaxation is smaller than the time of the mechanical action period 1/ω (< 1/ω) and the liquid has time to return to equilibrium state during this time 1/ω. At low temperatures, because of the high value of relaxation time, the condition ω > 1 is fulfilled and periodic changes ∆To do not return the system to the state of a metastable liquid. For the time of mechanical action 1/ω the liquid structure does not have time to relax to equilibrium. In the intermediate temperature range, when the temperature of the system decreases and corresponds to the condition ωg = 1, the system is trapped into a frozen state that does not return to the state of a metastable liquid as the thermodynamic temperature decreases.

3. Relaxation theory of Volkenshtein – Ptytsyn

Volkenshtein and Ptitsyn [6, 7] developed a rigorous physical theory, where the behavior of kinetic units that can be in two states with different energies separated by an energy barrier (see Figure 3.1).

Figure 3.1. The theoretical scheme of Wolkenstein-Ptitsyn [6]. 1 –ground state of the particle, 2 – its excited state. U1 and U2 – kinetic barriers for the 1 → 2 and 2 → 1, respectively, ΔE = (U1 - U2), the energy of the excited state is greater than the energy of the ground state by an amount ΔE.

The energy of excited state 2 with a fraction of particles n2 is greater than the energy of the ground state 1 with a fraction of particles n1 by the value ∆E = (U1 – U2), where U1 and U2 are the kinetic barriers for the 1 → 2 and 2 → 1 transitions, respectively. The kinetic equation for such a system has the form analogous to formula (2.2),

, (3.1)

where n20 is the equilibrium value of n2, is the relaxation time. The solution of equation (3.1) and the rather laborious mathematical analysis show that at a certain temperature Tg the value n2 of the system is frozen. The Tg corresponds to the condition

. (3.2)

From the theory it follows that the equilibrium balance of particles essentially changes only in a very narrow interval (range) of temperatures. This is manifested in the fact that the on cooling annihilation rates of active particles pass through a maximum and disappear in the course of time. The criterion of glass transition during cooling is the ratio obtained on the basis of a mathematical solution of the problem of finding the maximum of the function describing the freezing rate of the structure,

. (3.3)

Here is determined by authors as the temperature range characterizing the interval of transition from liquid to glass during the process of cooling.

From the theory of Volkenshtein-Ptitsyn [6, 7] there follow known experimental regularities of the glass transition process: the dependence of Tg on q, the breaks on the curves of the volume and enthalpy versus temperature at T=Tg. The theory describes hysteresis phenomena in the case of glass transition. At the same time, the theory does not give a quantitative agreement with experiment, since it does not take into account the presence of the spectrum of relaxation times and cooperativity of the process.

We note that relation (3.3) coincides with glass transition equations (1.1) and (2.7).

4. Configuron percolation theory

The glass transition is similar to a second-order phase transition in the Ehrenfest sense with continuity of volume and entropy, but discontinuous changes (kinks) of their derivatives. Structural rearrangements that occur in an amorphous material at glass transition lead to characteristic jumps of derivative thermodynamic parameters such as the thermal expansion coefficient (TEC). Because of that the glass transition has been considered as a second order like phase transition in which a supercooled melt yields, on cooling, a glassy structure and properties of an isotropic solid material. In experiment namely these discontinuities (kinks) allow to detect the Tg with typical heating rates used 3 – 5 K/s (see Figure 4.1.).

Figure 4.1. Temperature dependence of deviation of thermal expansion coefficient (TEC) from its value in the glassy state (T)=TEC-100.10-7 of a high-sodium borosilicate glass 60.SiO2.10.B2O3.3.Al2O3.20.5.ZrO2.20.Na2O designed for nuclear waste immobilisation near the glass transition temperature Tg=565 oC (given in units of.476.10-7 1/K).

The configuron percolation theory (CPT) treats glass transition as a threshold effect when percolation occurs via broken chemical bonds termed configurons. The glass transition temperature is given by CPT [24, 25, 32-34] as follows:

(4.1)

where Hd and Sd are quasi-equilibrium enthalpy (Hd) and entropy (Sd) of configurons. Here fc is the percolation threshold e.g. the critical fraction of space occupied by configurons at percolation. Melts such as SiO2 have the percolation threshold equal to the universal Scher-Zallen critical density fc=c = 0.15±0.01 whereas complex materials are characterised by material-dependent fc<< 1 [32-34]. The glass transition is however kinetically controlled and exhibits a range of Tg which depends on the cooling rate q with maximal Tg at the highest rates qmax. CPT explains the cooling rate dependence of Tg as due to configuron relaxation and decrease of percolation threshold with cooling rate [32]. Indeed breaking of bonds is always associated with strain-releasing local adjustments of centres of atomic vibration where the instantaneous breakage of a bond introduces a non-equilibrated configuron of initial volume Vd which will then gradually evolve to a relaxed volume Vc. This leads to the following dependence of glass transition temperature with cooling rate:

(4.2)

where q0 is a standard cooling rate of the order of 0.005 – 0.2 K/s, and m <1 is the exponent in the Rayleigh-Plesset equation for the dynamic of a bubble in a liquid e.g. m=2/3 for B2O3 [32]. Equation (4.2) shows that the CPT practically results in the Bartenev equation (1.8) which evidences on its fundamental aspect. The coefficients a1 and a2for the Bartenev-Ritland equation within CPT are therefore:

(4.3)

(4.4)

Comparison of (4.4) with (1.10) shows that the activation energy in (1.7) is:

(4.5)

Hence the activation energy of relaxation is higher than Hd as the Rayleigh-Plesset exponent is smaller than unit. Notable that both a1 and a2 are inverse proportional to Hd e.g. there is a linear correlation between a1 and a2 which conforms with experimental data (see Fig. 1.4).

5. Glass transition equation and typical values of qg

Nemilov [35] introduced a single denotation δTg for the right parts of equations (2.7) and 3.3) following classic theories of glass transition [6, 7, 41]

.

Then the relationship between the cooling rate q and the structural relaxation time g is expressed by the general relationship - the glass transition equation [35]

,(5.1)

which determines the appearance of a glassy state at a temperature Tg in the cooling process. According to Nemilov [35], "the right-hand side of equation (5.1) is a scale factor that introduces the value of the temperature change, which corresponds to a certain change in the relaxation time necessary for the appearance of the glass." This equation is the most important result that follows from the relaxation theories of Mandelstam-Leontovich [41] and Volkenstein-Ptitsyn [6, 7]. In the future, a comprehensive study of this relation is of interest. First of all, it is necessary to find methods for calculating the value of δTg – the "temperature range characterizing the transition interval from liquid to glass during cooling" [35]. Note that δTg is not the "macroscopic" glass transition interval ∆Tg, which is defined as the temperature range in the vicinity of Tg, where physical properties (volume, enthalpy, heat capacity, etc.) change sharply (see Figure 1a). From general considerations, it could be expected that δTg fits into ∆Tg, but their equality does not follow from anywhere. The glass transition temperature Tg is experimentally located as the coordinate of the point of intersection of the extrapolated temperature dependences of the properties of the glass and the glass-forming melt. At the same time, the exact establishment of the beginning and end (boundaries) of the transition interval ∆Tg remains not completely determined (Figure 1a).

Taking into account the dependence of Tg on the cooling rate q, a number of researchers [38-40, 45] suggested using the concept of the standard cooling rate

q = 3 K/min = 0.05 K/s,(5.2)

adopted in glass technology. In the dilatometry of glasses and polymers, in virtually all countries, as a rule, the same cooling rate is used (5.2). In view of quite weak (semi-logarithmic) dependence of Tg on q, small variations of q near the standard value do not particularly affect the value of Tg, with rare exceptions. When q is varied by a factor of 10, the glass transition temperature shifts only by a small amount of only ∆T = 0.03Tg [38, 45]. Therefore, it is generally believed that the vast majority of the available data on Tg refers to the standard cooling rate. In the present work we use experimental data obtained at the standard cooling (heating) rate. For other values of q, there are little data to be quoted. Note also that the above values (1.5) of the structural relaxation time of inorganic glasses g≈(1-2)∙102s refer to the standard cooling rate of the melt [38-40, 42]. Thus, with the standard values of the cooling rate q = 0.05 K/s and the relaxation time g≈(1-2)∙102s, the product qg – the left-hand side of the glass transition equation (5.1) – for inorganic (in particular, silicate) glasses in order of magnitude is practically

K. (5.3)

Therefore, for the temperature range δTg – for the right-hand side of the glass transition equation (5.1) – under standard conditions, we can expect approximately the same values: δTg≈(5-10) K.

6. Estimation of the temperature range δTg6.1. Bartenev's calculation method

In view of the absence of a generally accepted theoretical formula for the peculiar temperature dependence of the structural relaxation time (T) in the glass transition region, Volkenstein and Ptitsyn [6, 7] have confined themselves to the simplest case where it is assumed that this dependence is described in the first approximation by an usual exponential, e.g. by Frenkel equation (1.7). Substituting (Т) from (1.7) for the glass transition condition (3.2) when U = const, they obtained the relation [6, 7]

,(6.1)

coinciding with the glass transition equation (5.1). With this approximation, the parameter of equation (5.1) acquires the following physical meaning

. (6.2)

Bartenev [38, 45] used this relation to calculate the parameter С in his equation (1.1). Then С is replaced by δTg, since they are equal С = δTg. The ratio (RTg/U) was determined from the relaxation time equation (1.7) at T=Tg

,

where o ≈ 10-12s, g ≈ 102s, and R is the gas constant. From this, taking into account (6.2), there follows a method for calculating δTg using data of Tg’s.

. (6.3)

An estimate of this formula for inorganic silicate glasses (when roughly Tg 800 K) gives the following approximate value

,(6.4)

which is substantially higher than the product qg≈ (5-10) K (see (5.3)). In our opinion, the overestimated value of (6.4) is due to the assumption that the activation energy of the vitrification process (U = const) is constant when calculating data with formula (6.1). It is known that the value of U near Tg increases sharply [3-5]. It is easy to verify that replacing the above procedure with the temperature dependence of the activation energy U(T), in place of (6.1) one can obtain a glass transition equation in which the parameter δTg is determined by the expression [36]

. (6.5)

Here Ug is the activation energy of the glass transition process at the glass transition temperature. The derivative (dU/dT)Tg has a negative sign, since upon cooling (dT<0), the activation energy increases (dU>0). Therefore, the expression in square brackets is greater than unit which means that formula (6.2) overestimates data.

6.2. Calculation of δTg using the Williams-Landel-Ferry equation

As noted above, in the glass transition region, the simple exponential dependence (1.7) with U = const does not give correct data. Therefore, empirical equations are proposed that implicitly take into account the temperature dependence of the activation energy of glass transition process U(T). Among them, the WLF equation (Williams-Landel-Ferry) [57, 58] was widely used for the relaxation time (T) and viscosity η(T)

,. (6.6)

The justification of this equation is shown in many works for various amorphous substances [5, 57-61]. Substitution of (T) from the equation (6.6) into the glass transition condition (3.2) leads to the glass transition equation in the form [36]

, (6.7)

from which follows the formula for calculating δTg using empirical parameters of the WLF equation С1 and С2

. (6.8)

An estimate for this ratio for silicate glasses follows from Table 6.1 and it is in satisfactory agreement with the product qg (5.3).

(6.9)

Table 6.1.Calculation of temperature range δTg for silicate glasses and amorphous polymers.

Glass (content, mol. %)

Т12,

К

Т13,

K

δTg,К

(6.15)

δTg, К

(6.17)

К (6.8)

С1

С2,

К

τg,

с

Sodium-silicate glassesNa2O – SiO2 [49]

15 Na2O – 85 SiO2

819

790

29

13

12

36

430

239

20 Na2O – 80 SiO2

792

766

26

11

11

36

390

217

25 Na2O – 75 SiO2

769

745

24

10

10

35

355

202

30 Na2O – 70 SiO2

749

727

22

10

9

35

322

184

33 Na2O – 67 SiO2

738

717

21

9

9

35

304

174

35 Na2O – 65 SiO2

726

705

21

9

8

35

291

166

Window glass[19]

846

825

21

9

8

36

305

160

Poly-alkali silicate glass[76]

69.04 SiO2 · 30.96 Na2O

736

718

18

8

7

46

340

147

79.29 SiO2 · 12.97 Na2O · 7.75 Li2O

700

683

17

7

7

45

315

140

43.22SiO2 · 9.55Na2O · 47.23CsO

721

704

17

7

6

31

200

129

71.59 SiO2 · 24.4 Na2O · 4.01 Li2O

695

681

14

6

6

36

231

128

Amorphous polymers[58]

Polyisobutylene

-

202

6

-

2.7

38

104

54

Polyvinyl acetate

-

305

9

-

1.3

36

47

26

Polyvinylchloroacetate

-

296

9

-

1.0

40

40

20

Polymethyl acrylate

-

276

8

-

1.1

42

45

22

Polyurethane

-

238

7

-

0.9

36

33

18

Natural rubber

-

300

9

-

1.4

38

54

57

Methacrylate polymers

ethyl

-

335

10

-

1.6

40

65

32

n-butyl

-

300

9

-

2.5

39

97

50

n-octyl

-

253

8

-

2.9

37

107

58

Note: Composition of window glass (mass%) [40]: SiO2 – 72.7; CaO – 8.6; MgO – 3.4; Al2O3 – 1.3; Na2O – 13.6; K2O – 0.4; τg = C2 / qC1, C1иC2 – parameters of the Williams-Landel-Ferry equation, q = 0.05 K/s.

6.3. Nemilov calculation method

Taking the logarithm of the relaxation time (see equation (1.7)) and taking temperature derivative (dln τ/dT) at U=const and T=Tg , we have

.(6.10)

From this expression and (6.2) it follows the Volkenstein-Ptitsyn formula for the temperature range [6, 7]

.(6.11)

Nemilov [35] proposes the equality sign in this formula to be replaced by a sign of proportionality, since in the Volkenstein-Ptitsyn theory the spectrum of relaxation times is not considered, but only a single time . Further it is assumed that the proportionality coefficient should be universal and somewhat arbitrarily assumed to be ln10 = 2.3:

. (6.12)

In the transition from the natural to the decimal logarithm (ln=2.3log), this proportionality factor is reduced and, taking into account the relationship between the relaxation time τ(T) and the viscosity η(T), the relation (6.12) is rewritten as [35]

.(6.13)

The quantity of is taken to be equal to the temperature range in which the viscosity η(T) changes by an order of magnitude, from 1013 to 1012 Pas, and equality (6.13) is reduced to a convenient form for calculation

, (6.14)

where Т12 and Т13 are temperatures, corresponding to logarithms of viscosities log12=12 and log13=13, hence we get finally the Nemilov’s relation [35]

. (6.15)

Calculations for 13 silicate glasses [35] were done using this formula. Majority of them have the value of Tg close to 20 K. An average of the 13 values is

, (6.16)

which coincides with the Bartenev’s estimate (6.4) and is about 2 times of typical values of product qg(5-10) K, as well as same of Tg values are obtained using the WLF equation (6.9). For sodium silicate glasses, the Nemilov formula (6.15) gives values (Table 6.1) as high as Tg=(21-29) K.

Since Nemilov used the Volkenstein-Ptitsyn formula (6.11) derived at U=const, it might be assumed that overestimated values of Tg from (6.16) are explained by the same reason as in Bartenev assessments, namely by incorrectly assuming the constancy of the activation energy of the glass transition process. However, in contrast to formula (6.2), Nemilov used the derivative (6.13), which does not depend on whether the activation energy of the glass transition process changes with or not with temperature, since in the derivative (6.13) the dependence of (Т) (or (T)) in the interval ∆T is a continuous monotonic function[footnoteRef:3]. [3: Nemilov, private communication.]

In this connection, the question arises as to the cause of the discrepancy between the estimate Tg for Nemilov and the product qg. Let us consider one of the explanations.

6.4 Volkenstein-Ptitsyn calculation

It is known that in many cases the glass transition of a liquid is described quite satisfactory with the use of a single average relaxation time, by which the most probable relaxation time corresponds to the maximum of continuous spectrum of relaxation times, characterizing the glass transition of liquids and polymers [1-7, 39]. In relaxation spectrometry this discrete relaxation time is denoted by and serves as a characteristic of the -relaxation process [39, 40]. In this connection, it is of interest to calculate Tg directly using the Volkenstein-Ptitsyn formula (6.11), taking in it as a time of -relaxation process. In other words, it is proposed to repeat the Nemilov procedure for calculating without introduction of an empirical coefficient of proportionality 2.3. It is easy to see that this approach leads to the following values for a number of silicate glasses (Table 6.1)

, (6.17)

which are consistent with the product qg and with data for the same glasses obtained using the WLF equation (6.9) (Table 6.1). This suggests that the overestimated values by (6.16) may be due to the arbitrary introduction of empirical factor 2.3 into the Volkenstein-Ptitsyn formula (6.13). The search for correct methods of calculating Tg is actually at the initial stage [35,36]. In further investigations of glass transition equation (5.1) obviously the real reasons for the discrepancy in the estimates of Tg for different authors will be found out. According to the approximate estimate of Volkenstein-Ptitsyn [6,7], the temperature range Tg is several degrees, which is in agreement with the classical Simon’s notion [62] that the structure of the glass-forming melts is frozen in a very narrow temperature range including Tg. The Volkenstein-Ptitsyn theory also leads to this conclusion [6, 7].

7. Fragility and temperature range Tg

In recent decades, the notion of fragility which is typically determined by the temperature dependence of the viscosity near the Tg [63,64] has been proposed

.(7.1)

It is easy to see that substituting the dependence of (T) from the WLF equation (7.6) into this expression leads to the relation [65]

.

Taking into account Tg=C2/C1 at m=const in glasses of one class it results in a linear correlation between Tg and the glass transition temperature Tg

,(7.2)

which is confirmed by experimental data (Figures 7.1-7.3). The value of m is used to classify glasses.

Figure 7.1. Linear correlation between δTg and Tgfor sodium silicate glasses. Content Na2O, mol. %: 1 - 15, 2 - 20, 3 - 25, 4 - 30, 5 - 33, 6 - 35. Data from [49].

Figure 7.2. Linear correlation between δTg and Tgfor amorphous organic polymers. 1 - polyhexene-1, 2 - polyurethane, 3 - polyvinylchloroacetate, 4 -polymethylacrylate, 5 - polyvinyl acetate, 6 -natural rubber, 7 -methacrylate ethyl. Data from [36].

Figure 7.3. Correlation between δTg and Tg for metallic glasses. Data from [77, 78]. 1 -Fe83B17, 2 - Fe80P13C7, 3 - Fe41.5Ni41.5B17, 4 - Pd82Si18, 5 - Pd77.5Cu6Si16.5, 6 - Pd40Ni40P20.

8. Equation of glass transition in the model of delocalized atoms

Let us turn to the model of delocalized atoms [66], one of the important parameters of which is the fluctuation volume of the amorphous system ΔVe, which arises as a result of thermal displacements of particles from equilibrium positions

.

Here Ne is the number of delocalized atoms, Δve is the elementary fluctuation volume necessary for the delocalization of active atom, i.e. its displacement from the equilibrium position. The mobility of delocalized atoms in the glass transition region is determined mainly by the fraction of the fluctuation volume

.

From the point of view of this model, the parameters of the WLF equation (6.6) acquire the following physical meaning [66]

, ,

(8.1)

whereis the fraction of fluctuation volume frozen at the glass transition temperature, βf – coefficient of thermal expansion of fluctuation volume at T= Tg (). The product βfTg depends only on the value of fg [66,67]

.

(8.2)

From expressions (6.8), (8.1) and (8.2) it follows that the relative temperature interval (δTg/Tg) is a single-valued function of the fraction of fluctuating volume fg, frozen at the glass transition temperature,

.

(8.3)

Thus, in the framework of model considered the parameter of the glass transition equation δTg is determined by the Tg and the fraction of the fluctuation volume fg frozen at T=Tg [68]. It is noteworthy that the parameter of the WLF equation C1 is practically an universal constant, at least for glasses of the same class (Table 8.1). This means that the fraction of the fluctuation volume fg = 1/C1 is constant (see (8.1)) (Table 8.1)

.

(8.4)

Table 8.1. WLF equation parameters C1, C2 and the glass transition characteristics of amorphous substances

Amorphous

substance

,

К

,

К

,

K

К

(8.3)

τg,

с

C0

%

Sodium silicate glassNa2O–SiO2 [49]

Na2O, mol. %

15

782

36

430

12

0.028

7.8

6.1

240

0.5

3.5

20

759

36

390

11

0.028

7.8

5.9

220

0.5

3.4

25

739

35

355

10

0.028

7.8

5.8

200

0.5

3.5

30

721

35

322

9

0.028

7.8

5.6

180

0.5

3.5

33

712

35

304

9

0.028

7.8

5.6

180

0.6

3.5

35

705

35

291

8

0.028

7.8

5.5

160

0.6

3.5

Metallic glass (amorphous alloys) [77]

Pd40Ni40P20

602

39

93

2.4

0.026

7.1

4.3

48

0.8

3.6

Pt60Ni15P25

500

37

95

2.6

0.027

7.5

3.7

52

0.8

3.6

Pd77.5Cu6Si16.5

653

38

100

2.6

0.026

7.1

4.6

52

0.8

3.7

Fe80P13C7

736

38

120

3.2

0.026

7.1

5.2

64

0.8

3.6

Amorphous organic polymers [58] and selenium [5]

Polyvinyl acetate

305

36

47

1.3

0.028

7.8

2.4

61

0.8

3.5

Natural rubber

300

38

54

1.4

0.026

7.1

2.1

57

0.8

3.6

Methacrylate ethyl

335

40

65

1.6

0.025

6.8

2.3

50

0.8

3.6

Selenium

303

32

58

1.8

0.031

8.9

2.7

44

0.8

3.6

Low molecular weight organic glass [61]

Propanol

98

41

25

0.6

0.024

6.4

0.6

12

0.7

3.6

Prothylene Glycol

160

44

40

0.9

0.023

6.1

1.0

18

0.7

3.7

Glycerol

185

42

53

1.3

0.024

6.4

1.2

26

0.7

3.9

Note: fg=1/C1, τg=C2/C1q.

Linear correlation between δTg and Tg follows from the equation (8.3) at fg≈ const. Indeed, as noted above, a linear correlation is observed in a number of glasses between the Tg and the temperature interval δTg (determined from formula (6.8)) (see Figures 7.1-7.3).

Knowing fg and Tg, and using formula (8.3), we can estimate the value of the temperature interval δTg characterizing the transition of a liquid into a glass (Table 8.1) [68]. For sodium silicate glasses it is equal to δTg = (5 – 6) K, and for amorphous organic polymers it is δTg≈ 2 K. Glassy selenium (δTg =1.8K) is an inorganic amorphous polymer with a chain linear structure -Se-Se-Se-. Metallic glasses on the values of δTg≈ (4 – 5) K are close to silicate glasses than to amorphous polymers. In low-molecular organic glasses we have δTg≈ 1 K (Table 8.1).

Thus, the results of temperature interval calculation using the model of delocalized atoms of the temperature interval δTg≈ (5 – 6) K for sodium silicate glasses are in satisfactory agreement with the product qτg≈ (5 – 10) K for silicate glasses (5.3) and the result of calculating δTg from data on parameters of the VLF equation (6.9).

For most of the glassy systems the value of δTg according to (8.3) at fg≈0.025 is about 0.7% of the glass transition temperature and the relative temperature range (δTg/Tg) is practically universal for the amorphous substances studied (Table 8.1)

.

(8.5)

The fact that the temperature interval δTg turns out to be very narrow is explained by the low value of the fraction of the fluctuation volume fg frozen at the Tg, in other words, by the small scale of the local structure fluctuations near Tg. From the above methods for calculation of the temperature range δTg, in our opinion, the most preferable is the estimation from the data on the parameters of the WLF equation using formula (6.8). The calculation of δTg for amorphous organic polymers and low-molecular organic glasses from the data on C1 and C2, and also on the model of delocalized atoms (8.3) leads to low values (table 8.1)

,

which are consistent with the product qτg≈ (1 – 3) K at the standard cooling rate. Metallic glasses in terms of the valued of δTg ≈ (3 – 5) K occupy an intermediate position between inorganic and organic glasses (Table 8.1).

Since in the glasses of one class it is C1 ≈ const, a linear correlation between C2 and Tg is observed. Therefore, the equation (6.6) can be expressed in a modified form

,

where dimensionless value C0

,

unlike C2 depends weakly on the nature of the glasses: C0≈const (see Table 8.1).

Thus, the Williams-Landel-Ferry equation can be written in a modified form, where two dimensionless practically "universal" constants C1 and C0 appear.

In the Volkenstein-Ptitsyn relaxation theory of vitrification [6, 7], as noted above, molecules can be in two energy states in the liquid, namely, in the ground and excited states, but the physical meaning of the process of particle excitation – its transition from the ground state to the excited state – is not explained. Note that CPT briefly described above treats excitations as a bond breaking process [22, 24, 25]. Apparently, a delocalization of the active atom in the first approximation can be regarded as one of the possible variants of particle excitation by Volkenstein and Ptitsyn. Then the number of delocalized atoms corresponds to the number of excited (active) particles in the given theory. Under delocalization of an atom, for example, in silicate glasses is meant some critical transverse displacement of the bridge oxygen atom in the Si-O-Si bridge [66].

Concentration of delocalized (bridge) atoms (Ne/N) responsible above Tg for viscous flow in the glass transition range decreases to an insignificant value, of the order of (3-4)%, which is equivalent to "freezing" (Table 8.1),

,

where Δεe = piΔve is the energy of atom delocalization, equal in magnitude to the work of the atom displacement made against the internal pressure pi, due to the forces of interatomic (intermolecular) attraction. At Tg the process of atom delocalization (the transition of the particle from the ground state to the excited state) is frozen. In the process of glass softening, the process of atom delocalization is gradually unfrozen with heating and the number of delocalized atoms increases from small values in the frozen (glassy) state to their concentrations corresponding to the liquid state. In the model of delocalized atoms, in our opinion, the fraction of the fluctuation volume f, which is determined mainly by the concentration of delocalized atoms, acts as an internal (structural) parameter of the ξ type. From this point of view, the relative temperature interval (δTg/Tg) in (8.3) is a single-valued function of the internal parameter fg, characterizing the structure of the system near the Tg.

9. Kinetic criterion of glass transition

In a series of papers by Gutzow, Schmelzer and others [1, 2, 69], the following criterion of glass transition was formulated

, with .

(9.1)

As noted by Nemilov [35] in oxide (e.g. silicate) glasses at T=Tg the product of qτg according to (9.1) should correspond to the value Tg ≈ 800 K

,

(9.2)

which contradicts classic relaxation theories of glass transition [6,7,41,62]. Typical values of the product qτg are about qτg≈ (5-10) K [see (5.3)]. Using the results of the above study, let us consider one of the options for clarifying the glass transition criterion (9.1). If we divide both sides of the glass transition (5.1) by the glass transition temperature Tg, we arrive at the relation

,

(9.3)

where value of Cg is actually a universal constant [equations (8.3) and (8.5)]

.

(9.4)

The relation (9.3) can be considered as a refined version of the kinetic criterion of glass transition (9.1) and written it in a generalized form

,.

(9.5)

Suggested condition of glass transition (9.5) does not contradict classic relaxation theories [7,62]. For the product qτg applied to oxide glasses (Tg≈ 700 – 800 K) it gives the following value

,

which is in agreement with typical values for qτg.

10. Estimation of the fraction of fluctuation volume fg

As noted above, the main regularities of the glass transition process are qualitatively the same for all amorphous substances, regardless of their nature: for amorphous organic polymers, inorganic glasses, metallic amorphous alloys, aqueous solutions, chalcogenides, etc. Thus, the dependence of Tg on the cooling rate (q) is described by the same Bartenev equation (1.8), where the ratio of the empirical parameters (a2/a1) is practically constant for various amorphous substances, including organic amorphous polymers and silicate glasses (see Figure 1.4 and Table 1.1) (1.11). Attention is drawn to the coincidence of the values (a2/a1) and fg in the equalities (1.11) and (8.4). As can be seen below, this is not accidental. Taking into account expression (1.10), the Bartenev relation (1.8) can be represented in the form

.

(10.1)

If we compare the expression for viscosity [66]

(10.2)

and the well-known Frenkel equation (see [70])

,

written at T = Tg, we can trace the relationship between the activation energy of the viscous flow and the fraction of the fluctuation volume fg:

.

As a rule, the activation energy of glass transition and viscous flow coincide. Then relation (10.1) takes the form

.

(10.3)

Thus, it turns out that the fraction of the fluctuation volume fg frozen at Tg depends on the cooling rate of the melt, which is quite natural. At a given cooling rate q = const, the value of fg is also constant. Substituting in formula (10.3) the standard value of q from equation (5.2) and the quantities a1 and a2 from table 1.1, we come to the conclusion that the fraction of the fluctuation volume fg calculated from the data on the dependence of Tg on the cooling rate (Table 1.1)

,

(10.4)

is consistent with the results of calculating fg using viscosity in the glass transition region (8.4) (Table 8.1). The values of fg in equations (8.4) and (10.4) refer to the standard cooling rate. We see that the constant of the Williams-Landel-Ferry equation C1 = 1/fg is closely related to the ratio of the parameters of the Bartenev equation (a2/a1). As a rule, in the equality (10.3) (a1/a2)>>logq, therefore C1 and (a1/a2) practically coincide.

At the same time, the question arises why the dependence of Tg on the cooling rate (1.8), which is derived without taking into account the temperature dependence of the activation energy of the glass transition process U = U(T), is quite successful. To clarify this question, we consider the derivation of the Bartenev equation (1.8) with the account of the temperature dependence of the activation energy U(T).

11. The generalized Bartenev equation

In the model of delocalized atoms [66, 67], the fraction of the fluctuation volume fg frozen at Tg is expressed by the relation

, (11.1)

where v=V/N is atomic volume, Δve is an elementary volume, necessary for the atom delocalization, Δεe is the energy of atom delocalization. Substituting fg from this relation into the viscosity equation (10.2), we arrive at the expression

,

which coincides with the empirical equations of viscosity proposed by Bradbury [71], Shishkin [72] and Waterton [73] independently of each other.

Taking into account the well-known relationship between the relaxation time τ(T) and the viscosity η(T), for the temperature dependence τ(T) one can write the analogous expression

.

(11.2)

Here it is assumed that the atom delocalization volume is close to atomic volume (v/Δve1) [74]. Substituting into the glass transition equation (1.1) the relaxation time τg from (11.2), after some transformations we obtain the generalized Bartenev equation

,

(11.3)

where b1 = R/∆εe; b2 = ln(C/τ0); a1 = b1lnb2.

Thus, the dependence of the glass transition temperature on the cooling rate Tg(q) is weak, since q enters the resulting expression under the double logarithm. Assuming τ0≈ 10-12 s and С≈ 10 K [36] at the standard cooling rate, we can estimate b2 = ln(C/τ0) ≈ 30. At relatively low cooling rates lnq˂˂b2, the logarithm on the right-hand side of (11.3) can be expanded in a series and limited to its first member

Then the equation (11.3) becomes Bartenev's equation (1.8)

,

where a2 = b1/b2. Consequently, the Bartenev equation (1.8) is valid at not very high rates: lnq<< 30. Indeed, with an increase in the cooling rate for a number of glasses, for example, lead-silicate, a deviation from the dependence (1.8) is observed (figure 1.3) [45]. We note that the parameter С in the glass transition equation, although weakly, depends on the cooling rate q through Tg(q). However, Tg depends logarithmically on q and, in addition, С is under logarithm. Therefore, the parameter С can be regarded as practically constant, which is confirmed by the justification of the Bartenev-Rittland equation at moderate cooling rates.

To verify the obtained relation (11.3), graphs were plotted in coordinates 1/Tg – ln(1– (lnq/30)) for various amorphous substances [75]. The experimental data [38, 45, 49] in these coordinates lie on the straight lines (see, for example, Figures 11.1-11.3), including in lead-silicate glasses (Figure 11.3), in which a deviation from the Bartenev equation (1.8) (Figure 1.3) is observed.

Figure 11.1. Dependence of Tg on cooling rate for glassesAs-Sb-Se and Te-Gein the coordinates of equation (11.3): 1/Tg – ln[1 – (lnq/30)]. Data from [49]: (Mahadevan S., 1986 and Lasocka M., 1976). Parameters of equation (11.3) forAs-Sb-Se:a1 = 2,1 · 10-3K-1, b1 = 4,5 · 10-3K-1; and forTe-Ge: a1 = 2,4 · 10-3K-1, b1 = 5,6 · 10-3K-1

Figure 11.2. Dependence 1/Tg on ln(1 – lnq/30) for Se-Ga glass. Data from [49]: (El-Oyoun M.A., 2003). Parameters of equation (11.3): a1 = 3 · 10-3K-1, b1 = 5,7 · 10-3K-1.

Figure 11.3. Dependence of the glass transition temperature on the heating rate for lead silicate glasses №1 and №2 in coordinates 1/Tg – ln(1 – lnq/30). Data from [45]. Parameters of equation (11.3) for glass №1: a1 = 1,3 · 10-3K-1, b1 = 7,4 · 10-3K-1; and glass №2: a1 = 1,2 · 10-3K-1, b1 = 6,2 · 10-3K-1.

Thus, the resulting generalized equation (11.3) agrees satisfactorily with the experimental data over a wider range of the cooling rate than the Bartenev equation (1.8) [75].

12. Temperature dependence of viscosity of glass-forming melts 12.1. Empirical equations of viscosity of glass-forming liquids

It is known that, in contrast to simple liquids like acetone, the temperature dependence of the viscosity η(Т) of the glass-forming melts is not described by the usual Arrhenius exponential formula. In view of the absence for them of the generally accepted theoretical dependence η(Т), attempts were repeatedly made to establish which of the known empirical relationships best describes the temperature dependence of the viscosity.

Mauro et al. [79], from comparison the equations of Waterton [73], Vogel-Fulcher-Tamman [80-82] and Avramov-Milchev [83], containing three adjustable parameters, give preference to the Waterton formula ("double exponential"),

. (12.1)

Bradbury et al. [71] and Shishkin [72] proposed an analogous equation

. (12.2)

Pospelov [84], having analyzed a number of viscosity equations, also came to the conclusion that the "double exponential" type (12.1) is the more preferable of them. Since in the Waterton relation (12.1) the exponential dependence in square brackets is a strong function of temperature in comparison with the pre-exponent (B1/T), the latter can be taken practically for a constant. Therefore the formulas of Waterton (12.1) and Bradbury-Shishkin (12.2) actually coincide.

Meerlender [85] compared the five most common empirical equations of viscosity, including the relations (12.1) and (12.2), and came to the conclusion that the experimental data in a wide temperature range is more accurately described by the Enckel equation [86] if a small change is introduced and written in a generalized form [85],

. (12.3)

At low temperatures, in the glass transition region, the first term (B/T) in square brackets of this equation can be neglected in comparison with the exponent. Then it transforms into the Waterton relation (12.1). In turn, the well-known Williams-Landel-Ferry equation [57, 58] which is equivalent to the Vogel-Fulcher-Tamman relation (see [5, p. 90, 91]) is alogebraically derived from the "double exponential" formula types of the Waterton (12.1) and Bradbury-Shishkin (12.2).

Thus, the Enckel equation (12.3) can be regarded as one of the generalized variants of the main empirical relationships for the viscosity of glass-forming melts, and therefore it is of interest to study its nature.

The temperature dependence of the viscosity of glass-forming liquids according to Enckel [65] is due to the processes of dissociation and association of molecules. It was assumed that with a change in temperature, the degree of association of particles in these systems varies, which is accompanied by the rupture and restoration of interatomic and intermolecular bonds in the vicinity of this kinetic unit. From these representations, Enckel obtained the viscosity equation, which we write down as it appears in the original [76],

, (12.4)

where the expression in square brackets reflects the change in the degree of association of molecules upon cooling and heating of the liquid.

As we see, in the Enckel theory only one energy quantity Q is considered, which is related to the energy of decay of molecular associates. For liquids such as glucose and alcohols, Q is close to the energy of the hydrogen bond.

The generalized version of the Enckel equation (12.3) proposed by Meerlander [85], unlike the original (12.4), contains two energy values B and D. The physical significance of these activation parameters RB and RD (R is the gas constant) was not discussed by Meerlander. He only showed that in the generalized form the Enckel equation is in better agreement with experiment.

The possible variants of justifying this equation were considered [5, 87-89], different from the approach of Enckel [86].

Sanditova and Munkueva [74] propose an updated derivation of the Enckel equation, from which it follows that the given formula (12.3) actually contains not four, but three adjustable parameters (A, B and D).

Finally we note that the CPT gives for the shear viscosity coefficient temperature dependence a universal equation valid both for glasses and melts [11, 24, 25, 90-96]:

(12.5)

where A1, A2, and C are constants. Comparison with other viscosity models and numerical calculations using a genetic code based fitting procedure have confirmed the well-known (see [97]) excellent description of the viscosity by equation (12.5) for various glass systems both simple and complex organic and inorganic compositions [93]. From equation (12.5) it follows that the thermodynamic parameters of bonds can be found from available experimental data on viscosity [24].

12.2. Derivation of viscosity equation

Using the concept of a two-stage elementary flow of glass and their melts [98-100], we assume that the probability of the transition of the kinetic unit W from one equilibrium position to another is determined, firstly, by the probability of a local configurational change in the structure of W1 for a given kinetic unit and secondly, by the probability W2 that it has sufficient energy to jump into a new position (in the micro-region of the structural change)

W = W1·W2.

In this case, the local configurational change in the structure of the kinetic unit precedes its jump and is considered as a necessary condition for the realization of the latter (similar to the way for the atom to be able to move to a neighboring position a hole near it should be formed in advance where the atom can jump [98, 99]).

According to Muller [100], a preliminary local low-activation deformation of the structural network of interatomic bonds serves as a necessary condition for switching the valence bond, the basic molecular mechanism of the viscous flow of inorganic glasses and their melts. The latter can be adopted as one of the variants of the local configurational change in the structure of the kinetic unit responsible for yield.

At elevated temperatures, molecular mobility is determined by the probability of a hopping of the kinetic unit into one of the "readily loosened (deformed) microregions", which, as a rule, exist at high temperatures (W1 = 1),

. (12.6)

With temperature decrease, in the glass transition region, the structure of the melt is compacted and the probability of a local change in the structure of W1 plays an important role in the mobility of molecules. Let us consider this concept with the use of a model of delocalized atoms [66, 101].

Delocalization of the bound atom – its displacement from the equilibrium position – in amorphous substances is accompanied by the rearrangement of neighboring particles and, in fact, reflects the local configurational structural change. Taking this into account, by probability W1 we mean the probability of delocalization of an atom, which is expressed by the formula [66],

, (12.7)

where Δve is an elementary fluctuation volume necessary for delocalization of an atom, is the average fluctuation volume per kinetic unit. The fluctuation volume of an amorphous substance ΔVe arises as a result of delocalization of atoms – their thermal displacements from the equilibrium position (Ne is the number of delocalized atoms): ΔVe = NeΔve (see section 8).

Using the quasilattice model [102], we reveal the explicit form of the temperature dependence of the fluctuation volume vf(T). Suppose that when the bound atom is delocalized as a result of the local deformation of the lattice (bond network), the corresponding node is displaced from the basic position. Such a defective shifted lattice site will be called an excited node. Their number is equal to the number of delocalized atoms (Ne). The total number of nodes in the lattice, both the ground (N) and the excited (Ne), is (N+ Ne).

Taking this into account, for the free energy component of the system associated with the presence of lattice defects, we can write relation

,

where the expression under the logarithm reflects the number of ways of placing delocalized atoms at their possible nodes, is the energy of delocalization of the atom.

It can be shown that the minimum ΔF corresponds to the following number of delocalized atoms

. (12.8)

The fraction of the fluctuation volume

taking (12.8) is defined by the formula ()

. (12.9)

Substituting this dependence f(T) in the exponent of the equation (12.7),

,

we arrive at an expression for the probability of a local change in the structure of W1(T) in the form of a "double exponential"

.(12.10)

Further, using the relations (12.6) and (12.10), and also the known relation between the viscosity η and the value of W in accordance with the Stokes-Einstein theory (see, for example, [5]),

,

we finally obtain the equation of viscosity

,(12.11)

which practually coincides with the empirical equation of Enckel (12.3). Here η0 is the proportionality coefficient (viscosity η at T → ∞). Apparently, we can assume [87] that the proportionality coefficient η0 coincides with the pre-exponent of the well-known Eyring equation η0= h/vη, where vη is the volume of the particle overcoming the potential barrier, h is the Planck constant.

12.3. Comparison with the experiment

There are three parameters in the equation (12.11): η0, ∆F∞ and ∆εе. The pre-exponential factor η0 corresponding to the viscosity at elevated temperatures is found by extrapolating (using the Lagrange polynomial [82]) the viscosity curve log η– 1/T to the temperature Т →∞ [104]. Knowing the value of η0, by adjusting the remaining two parameters ∆F∞ and ∆εе the viscosity η can be calculated from equation (12.11).

Experimental data on the temperature dependence of the viscosity of the glass-forming melts η(Т), necessary for comparison with the results of the calculation, were taken from the SciGlass database [49]. Two-component silicate, germanate and borate glasses were chosen as research objects. On the graphs in the coordinates log η- 1/T the points correspond to the experimental data, and the solid curves are calculated by the equation (12.11). As can be seen, the theoretical curve falls well on the experimental points (Figures 12.1, 12.2 a, b, c, d).

Figure 12.1. Temperature dependence of the viscosity of potassium silicate glasses. Points - experimental data, curves - calculation by equation (12.11).

Na2O-SiO2 a

PbO-SiO2 b

33.3

Li2O-SiO2 c

Na2O-GeO2 d

Figure 12.2. Temperature dependence of the viscosity of sodium silicate (а), lead silicate (b), lithium silicate (c) and sodium germanate (d) glasses. Points - experimental data, curves - calculation by the equation (12.11).

Table 12.1 gives the values of the parameters η0, ∆F∞ and ∆εе for which the calculation is consistent with the experiment.

Table 12.1. Characteristics of viscous flow of inorganic glasses R2O-SiO2 (R = Li, Na, K), PbO-SiO2, Na2O-GeO2 and Na2O-B2O3

R2O (R=Li, Na, K),

PbO,

mol.%

- log η0 [П]

Δεe,

kJ/mole

ΔF∞,

kJ/mole

Tg,

К

ΔFη(Tg),kJ/mole

fg

Δεe,

kJ/mole

(13.8)

Li2O

Li2O-SiO2

10

14

25

30

33,3

2.55

2.57

2.41

2.25

2.23

20

20

19

19

19

127

120

91

78

71

814

788

738

721

708

245

237

219

212

208

0.028

0.028

0.028

0.028

0.028

24

23

22

21

21

Na2O

Na2O-SiO2

15

20

25

30

33

2.53

2.35

2.36

2.36

2.26

19

19

19

19

19

118

101

94

87

78

783

759

739

721

712

235

225

219

214

209

0.028

0.028

0.028

0.028

0.028

23

23

22

21

21

K2O

K2O-SiO2

13

15

20

25

2.33

2.31

2.14

2.22

19

19

19

19

121

117

100

90

795

793

759

739

235

232

222

217

0.028

0.028

0.025

0.025

24

24

23

23

PbO

PbO-SiO2

25

30

45

50

3.15

2.95

3.34

3.33

21

21

20

19

100

79

57

42

785

761

696

674

245

234

218

212

0.027

0.027

0.027

0.027

24

23

21

20

Na2O

Na2O-Ge2O

15

20

25

30

3.81

3.21

3.10

3.12

22

22

21

21

79

58

49

40

801

773

749

727

259

241

232

225

0.026

0.027

0.027

0.027

24

23

22

22

Na2O

Na2O-B2O3

10

15

20

25

30

3.45

3.93

3.41

3.77

4.45

18

20

21

22

23

47

44

26

19

3

618

680

727

735

748

194

221

229

237

250

0.026

0.026

0.026

0.026

0.025

19

21

22

22

23

Thus, the viscosity equation (12.11), derived from the refined model [88, 89], is in satisfactory agreement with the experimental data for glass-forming melts [74].

13. Mechanism of viscous flow and liquid glass transition

Viscosity is a principally important property, determining specifics of glassy state. Namely, increased viscosity, slowing the crystallization process, facilitates the transition of the melt into glass. Atomic mechanism of viscous flow in inorganic glasses is described in works of Douglas [105] and others [56, 106-108]. Muller [107] was the first to suggest the mechanism based on the activated switch of bridge bonds, i.e on the exchange of bridge atoms locations (Fig. 13.1).

Smyth, Finlayson and Remde [106] developed a model of viscous flow of silicate glass, which provides a possibility of the Si-O bond break and switch of Si-O bonds to the unsaturated bonds of silicon and oxygen (Figure 13.1).

1

2

3

4

1

2

3

4

- O

- Si

a

b

Figure 13.1. Scheme of bond switch in the theory of viscous flow of Smyth, Finlayson and Remde [106].

If the bridge ion of oxygen 2 as a result of lateral thermal vibrations approaches closely to unsaturated silicon ion 1, then a switch of valence bond can occur 2 – 3 to 2 – 1 and the change of partners, leading to new configuration of glass network. Authors [106] think, that main role in viscous flow is played by bonds switch to unsaturated silicon bonds, and not to oxygen bonds, as in accordance with Philipovich [108] and Sanditov [109] (Figure 13.2).

Figure 13.2. Scheme of viscous flow mechanism of silicate glasses, based on the assumption of the presence of oxygen ions with vacant bond [109].

2

3

1

2

3

1

2

3

1

a)

b)

c)

- O

- Si

F

Figure 13.3 shows the scheme of valent bridge bond switch to unsaturated silicon ion according to Nemilov [110].

Figure 13.3. Scheme of valent bridge bond switch in silicate glass according to Nemilov [110].

Notion was developed, that in this scheme the displacement of oxygen bridge atom in Si-O-Si bridge in the process of viscous flow of silicate glasses is composed of two steps [111]. The first step is its delocalization (displacement), leading to local low activated stretching of silicon-oxygen network (transition А → В). The second step is related to rise of bridge oxygen atom to the top of the potential barrier, which is corresponding to switch of valent bridge atom (transition В → С) – jump of bridge atom. Wherein first step А – В is considered as a necessary condition for the realization of the second step В – С. In the aforementioned theory (model) of the viscous flow probabilities of these transitions А – В and В – С are described by relations (12.7) and (12.6), respectively.

From comparison of the derived viscosity equation (12.11) with the known Eyring equation (see [56])

(13.1)

it follows, that free energy of activation of viscous flow ΔFη(T) is a sum of two summands

,(13.2)

where ΔF∞ can be named as a potential for hopping of kinetic unit (bridge atom), and ΔFS(T) – potential of local configurational change of structure, which is a function of temperature

.(13.3)

Value of ΔF∞ is a high temperature limit of free energy of yield activation, since at RT>>Δεe the second term in equation (13.2) becomes zero and viscosity equation (12.11) becomes a usual exponential dependence with constant free energy of activation

.(13.4)

At low temperatures in the glass transition region, atom delocalization energy Δεe becomes comparable with energy of thermal vibrations of the lattice (~3kT) and the number of delocalized atoms responsible for yield significantly decreases (according to law exp(–Δεe/kT)). Thus for activation hopping of bridge atom (valence bond switch) a preliminary local deformation of structural network (bridge atom delocalization, transition A → В, see Figure 13.3) is necessary: potential of local configurational change of structure ΔFS(T) sharply increases. This explains exponential growth of free energy of yield activation in the glass transition region.

Without delocalization of bridge atom (“trigger mechanism” of yield) it is impossible for a kinetic unit to hop and therefore for the viscous flow to occur. Thus, freezing of the process of active atom delocalization at low temperatures leads to cease of the viscous flow and transition of the melt into glassy state. This moment happens when the energy of thermal vibrations of the lattice (), reduced to one atom, becomes equal or less the atom delocalization energy:

. (13.5)

Number of degrees of freedom of the solid state I for a rough approximation is taken equal to i ≈ 6 (by analogy with ionic cubic crystal). As expected, energy Δεe determined independently from the data on the empirical parameter D of the Enckel equation (12.3),

,(13.6)

linearly depends on the glass transition temperature Tg (Figure 13.4).

Figure 13.4. Linear correlation between atom delocalization energy (Δεe = RD) and glass transition temperature (Tg) of sodium-silicate glasses Na2O-SiO2. Content of Na2O, mol. %: 15(1); 20(2); 25(3); 30(4).

Equation (13.6) is obtained from the comparison of relation (12.11) with empirical Enckel equation (12.3).

Calculation of atom delocalization energy for silicate glass №15 (D= 2500 К) [85]

(13.7)

is in satisfactory accordance with results of Δεe calculation from the model of delocalized atoms [66] for a number of silicate glasses (table 12.1)

. (13.8)

From this formula (13.8) also a linear correlation between Δεe and Tg follows, since fg≈const≈ 0.024 – 0.028 (Tables 1.1 and 8.1). In accordance with equations (13.5) and (13.8) the value of ln(1/fg) has to be close to three (i/2≈ 3). Indeed, at average value of fg≈0.025– 0.030 (Tables 1.1 and 8.1) we have

.

Value of Δεe calculated using formula (13.8) is constant in a wide temperature range in the glass transition region [112].

Thus, local configurational change of structure, which is described in the framework of the model of delocalized atoms, serves as a necessary condition for realization of elementary act of viscous flow of glass-forming liquids and is responsible for the peculiar temperature dependence of viscosity of glasses and their melts. Second summands in the viscosity equations (12.11) and free activation energy of yield (13.2), related to configurational structural change (atom delocalization process) represent basic difference between glass-forming melts and simple liquids like metal melts. At high temperatures (kT>>Δεe) they disappear and these equations transform into typical relations for simple liquids.

Conclusion

Glass transition of liquid has a pronounced relaxation character and obeys kinetic laws. Upon reaching the liquid-glass transition molecular rearrangements in the glass-forming melts become so slow, that structural changes cannot follow temperature changes. From relaxation point of view in the process of glass transition a decisive role is played by the relation between structural relaxation time τ and cooling rate of melt q = dT/dt. Interrelation of these values is expressed by glass transition equation: qτg = δTg, where τg is relaxation time at T = Tg, and δTg is the temperature interval, characterizing the transition range from liquid to glass during the process of cooling.

Calculation of temperature interval δTg using the data on parameters of Williams-Landel-Ferry equation and the model of delocalized atoms for inorganic oxide glasses is in an agreement with the product qτg≈ (5 – 10) K. An analogous statement is true for organic glasses, whose δTg have lower values δTg≈(1 – 3) K, consistent with the product qτg for these systems. We have used in this work experimental data related to standard cooling rate of the melt q = 0.05 K/s noting that there are few data obtained at other cooling rates. Interpretation of the glass transition equation was considered in the framework of the model of delocalized atoms and relaxation theory of glass transition. Derivation of Bartenev equation (1.8) with the account of the temperature dependence of the activation energy of glass transition process is proposed. It was shown, that Bartenev equation is true at relatively low cooling rates. Fraction of the fluctuation volume fg, calculated from the data on the dependence of Tg on cooling rate of the melt coincides with calculation of fg from the data on viscosity near the glass transition region. A refined version of kinetic criterion of glass transition is proposed. Frequency calculation was done equivalent to standard cooling rate of the melt. Only at low frequencies of mechanical action, approximately 10-3Hz, the temperature of dynamic glass transition in inorganic glasses coincides with Tg– simultaneous structural and dynamic glass transition occurs (e.g. topologic or structural and viscous components of deformation freeze at the same time). The model used should be viewed as a first step in the liquid-liquid transformation at Tg because the events at the glass transition are not limited to solid-like freezing while the solid-like state can be further transformed in an ultra-stable glass state [18].

Excited delocalized atoms (equivalent to broken bonds termed configurons in the CPT [96]) are responsible for viscous flow of glass-forming liquids, their concentration during cooling decreases and in the glass transition region reaches negligibly small values (around 3%), which is equivalent to freezing. At the glass transition temperature the process of atom delocalization is frozen (e.g. transition from ground to excited state).

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