# Interpretation of the apparent activation energy of glass transition

Koun Shirai  The Institute of Scientific and Industrial Research, Osaka
University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

###### Abstract

The temperature dependence of the viscosity of glass is a major concern in the
field of glass research. Strong deviations from the Arrhenius law make the
interpretation of the activation energy difficult. In the present study, a
reasonable interpretation of the apparent activation energy is demonstrated
along similar lines as those adopted in solid-state physics and chemistry. In
contrast to the widely held view that phase transition in glass occurs at the
reference temperature $T_{0}$ according to the Vogel–Fulcher–Tammann formula,
in the present work the transition observed at the glass-transition
temperature $T_{g}$ is regarded as a phase transition from the liquid to solid
phases. A distinct feature of glass is that the energy barrier significantly
changes in the transition range with width $\Delta T_{g}$. This change in the
energy barrier alters the manner in which the apparent activation energy
constitutes the Arrhenius form. Analysis of available experimental data showed
that the actual energy barrier is significantly smaller than the apparent
activation energy, and importantly, the values obtained were in the reasonable
range of energy expected for chemical bonds. The overestimation of the
apparent activation energy depends on the ratio $T_{g}/\Delta T_{g}$, which
explains the existence of two types of glasses strong and fragile glasses. The
fragility can be re-interpreted as an indication of the degree of increase in
the energy barrier when approaching $T_{g}$ from high temperatures. Since no
divergence in viscosity was observed below $T_{g}$, it is unlikely that a
transition occurs at $T_{0}$.

##  I Introduction

The core problem in glass research at present is what is the nature of the
glass transition Angell et al. (2000); Lubchenko and Wolynes (2005); Berthier
and Biroli (2011); Stillinger and Debenedetti (2013). Grass transition is
observed, for example, from specific heat ($C_{p}$) versus temperature ($T$)
curves. The transition temperature $T_{g}$ is the temperature at which the
specific heat exhibits a quick change. It can be experimentally observed, but
its value varies to some extent depending on experimental conditions such as
the cooling rate. This gives rise to the believe that $T_{g}$ is not an
intrinsic property of glass. Unlike the usual phase transitions, structural
relaxation plays a crucial role in glass transition: the status of experiments
on relaxation is reported in review papers Ediger (2017); Niss and Hecksher
(2018); Royall et al. (2018). The structural relaxation is reflected by
viscosity $\eta$. The viscosity of glass-forming liquid increases drastically
by more than ten orders of magnitude when the liquid temperature approaches
$T_{g}$ from a higher temperature (Fig. 1 (a)). The $T$ dependence of
viscosity is most commonly expressed by the Vogel–Fulcher–Tammann (VFT)
formula Vogel (1921); Fulcher (1925); Tammann and Hesse (1926),

$\eta(T)=C\exp\left(\frac{D}{T-T_{0}}\right),$ (1)

where $C$, $D$, and $T_{0}$ are material-dependent constants. The reference
temperature $T_{0}$ is less than $T_{g}$ and is generally $0.6$ to $0.8$ times
$T_{g}$ Angell (1997). While $T_{0}$ is obtained by extrapolation, many
researchers have attempted to identify its significance as the genuine
transition temperature, which is an intrinsic property of a specific glass.
The relation of $T_{0}$ with the so-called Kauzmann temperature $T_{K}$ is
being actively debated Ngai and Yamamuro (1999); Ito et al. (1999); Martinez
and Angell (2001); Sastry (2001); Tanaka (2003); Wang et al. (2006); Hecksher
et al. (2008). For normal liquids, the nature of viscosity is well understood
as a thermally activated process, which obeys the Arrhenius law,

$\eta(T)=\eta_{0}\exp\left(\frac{Q_{a}}{k_{\rm B}T}\right),$ (2)

where $k_{\rm B}$ is Boltzmann’s constant, and $\eta_{0}$ is a material-
dependent constant. In this case, $Q_{a}$ is clearly the activation energy.
This formula was derived from first principles Glasstone et al. (1964). For
normal liquids, the $Q_{a}$ values are of the order of a few tenths of
electron volts or less Ewell and Eyring (1937), which is a reasonable range
considering the chemical energies of materials. If this standard formula,
i.e., Eq. (2) is applied to the viscosity of glasses in a narrow range of
temperature, $Q_{a}$ values are found to be surprisingly large more than 1 eV
(in some cases, 10 eV) even for organic glasses whose melting temperature
$T_{m}$ is less than room temperature Davies and Jones (1953a, b); Hodge
(1994). There is no convincing theory to interpret the large values of
$Q_{a}$. Furthermore, the pre-exponential factor $A=1/\eta_{0}$ is extremely
large: typically, the magnitude is of the order of $e^{100}$ Hodge (1994).
Thus far, no theory that accounts for this extraordinarily large value has
been established.

At present, the meaning of activation energy of glass is not being actively
researched. Instead, by normalizing $T$ with $T_{g}$ in the Arrhenius plot,
which is now known as the Angell plot, researchers are attempting to discover
something universal for glass transition Nemilov (1995); Laughlin and Uhlmann
(1972); Angell and Sichina (1976); Angell (1988); not . In the normalized form
of the Angell plot, how largely the $T$ dependence of $\eta$ deviates from the
Arrhenius law is the main information that can be obtained; fragility is a
parameter used to quantify the degree of the deviation. Depending on the
fragility, glasses are classified as strong or fragile. Although this manner
of plot is useful for classifying glasses, it does not aid the chemical
interpretation of the activation energy, which is the basic element for
understanding properties of materials. Without a pertinent interpretation of
the activation energy, one cannot understand why the viscosity of glass
changes so largely when approaching $T_{g}$. It is necessary to appropriately
interpret $Q_{a}$ obtained from the Arrhenius plot: this is the aim of this
study.

A resolution of the problem of the activation energy of glass was achieved in
a very different context, namely, in terms of the state variables of glass.
Today, the standard view is that glass states are nonequilibrium states.
Although the glass transition is a nonequilibrium phenomenon—transition itself
represents nonequilibrium—unduly emphasizing the nonequilibrium character
hides the thermodynamic nature of glass. A frequently-claimed reason for the
nonequilibrium character is that the state of glass is not determined by
temperature and pressure ($p$) alone but is affected by the previous history
of treatments. Thermodynamics states are to be specified solely by the current
values of state variables; if the glass states were equilibrium, the
properties must be completely determined solely by $T$ and $p$, but this is
not true, from which the above conclusion is derived. However, the attribution
of this discrepancy to nonequilibrium state is an easy escape from the
difficulty. Careful observation reveals that this dependence of the past
treatment that the solid underwent is a common property of solids: any solid
has, to a certain degree, the dependence of its properties on the preparation
conditions. The fundamental question then is that besides $T$ and $p$ which
could be the other state variables of a solid: the term thermodynamic
coordinate (TC) is used to refer to state variable. This question was at times
considered by several authors to be a flaw of the theory Bridgman (1950);
Kestine and Rice (1970); Maugin and Muschik (1994) but was not solved until
recently. The author examined this topic by reappraising the definition of
equilibrium and TCs for solids, and defined in a consistent manner. In short,
the answer to the question of TCs for a solid is the time-averaged positions
$\bar{\bf R}_{j}$ of all the constituent atoms of the solid: note that
instantaneous positions are irrelevant, and only equilibrium positions have
sense Sta ; Shirai (2020). This consideration can also be true for the state
during glass transition, provided some restrictions are imposed. The thermal
response of solids is so fast—typically 10 ps—that the fast vibrational motion
is adiabatically decoupled from slow changes in the structure. This behavior
is expressed as the adiabatic approximation of the second kind Shirai (2020).
Then, the instantaneous state at timescales much longer than this response
time can be regarded as an equilibrium state, i.e., temporal equilibrium
Shirai (2020). The states in glass transition are states of temporal
equilibrium. Even during such a transition, all the averaged positions
$\bar{\bf R}_{j}(t)$ as well as the time-dependent temperature $T(t)$ are well
defined, and accordingly, the glass state is expressed by the instantaneous
values of all TCs.

A striking consequence of this conclusion is that energy barrier $E_{b}$ of
the structural relaxation of glass during glass transition is determined
solely by the present positions of atoms, $E_{b}=E_{b}(\left\\{\bar{\bf
R}_{j}\right\\})$, irrespective of the previous history. Since during the
transition, the structure is mainly determined by temperature, the structural
dependence of $E_{b}(\left\\{{\bf R}_{j}\right\\})$ can be represented as the
temperature dependence of energy barrier $E_{b}(T)$. The Arrhenius analysis
presumes that $Q_{a}$ is constant: however, this assumption does not hold for
glass transition. Hence, the apparent activation energies obtained from the
Arrhenius plot results in extraordinarily large values in the usual sense of
the term “chemical bond” Shi .

In this paper, the above conclusion regarding the temperature dependence of
$E_{b}(T)$ is validated by analyzing experimental data on viscosity. Further,
a pertinent interpretation of the apparent activation energy is provided. In a
previous study Shi , this interpretation was derived from an analysis of the
hysteresis in the $C$-$T$ curve. Viscosity measurement is a more direct method
to obtain the activation energy. The rest of this paper is organized as
follows: the theoretical basis for the present analyses is given in Sec. II.
Then, Sec. III shows the first evidence for the proposed interpretation based
on the relationship between the width of glass transition and the apparent
activation energy. The second evidence is obtained by directly evaluating the
temperature-dependent energy barrier: this evidence is described in Sec. IV,
which provides three examples of classes of glasses. An important implication
of the present results to glass transition is discussed in Sec. V. Section VI
concludes the study. Throughout this paper, the numerical values of viscosity
$\eta$ are presented in terms $\log\eta$(poise) with base of 10.

##  II Theoretical backgrounds

###  II.1 Glass state and glass transition

There exists a large gap in comprehension of the glass state between glass
physics and other areas of solid-state physics. For non-experts in the field
of glass research, glass is almost evidently a solid. The glass transition
that occurs at $T_{g}$ is a phase transition between the solid and liquid
phases. However, glass researchers do not think so. Many of glass researchers
consider glass as a special type of liquid, specifically, kinetically frozen
liquid. Hence, it is necessary at the outset to state the present view and to
explain the differences from the viewpoint of glass literature.

The signature of the glass transition is observed in a quick change in the
properties of glass, such as specific heat and thermal expansion, in a narrow
range of temperature Kauzmann (1948). Here, the range is specified by
$T_{g,1}$ and $T_{g,2}$ with its width $\Delta T_{g}=T_{g,2}-T_{g,1}$, as
shown in Fig. 1(b). In this paper, the term the glass state is restricted to
refer to the phase of a glass substance below $T_{g,1}$, and is regarded as a
solid phase. Above $T_{g,2}$ and below $T_{m}$, the state of the glass
substance is a supercooled-liquid state; there $C_{p}$ obeys almost the
classical limit of Dulong-Petitte law, $3R$ ($R$ is the gas constant) Angell
(1995). The state between $T_{g,1}$ and $T_{g,2}$ is referred to as the
transition state. The $C_{p}$-$T$ curve often exhibits complicated shapes
depending on the preparation conditions, which makes it difficult to identify
the width $\Delta T_{g}$. Usually, an averaging value is used for $T_{g}$,
unless special interest is paid for the width. In spite of the preparation
dependence, it is meaningful to determine the unique value for $T_{g}$ within
a reasonable range, provided suitable conditions are used Mazurin (2007);
Moynihan et al. (1976). In the relationship between the viscosity $\eta$ and
$T$, in contrast, there is no characteristic temperature other than $T_{0}$
when $\eta(T)$ is recasted to the VFT form. However, it will be shown later
that this is not true: see Sec. IV.1. Operationally, the $T_{g}$ can be
defined as the temperature at which $\log\eta=13$. Many of glass researchers
do not think that there is a sharp boundary between the supercooled-liquid
state and the glass state. Even in the glass state, atoms are migrating with
extremely slow velocities. From this view, $T_{g}$ cannot be the temperature
of a phase transition. Instead, $T_{0}$ in Eq. (1) deserves the genuine
transition temperature, below which the so-called ideal glass appears. The
reason that the ideal glass has not been observed is attributed to its
extremely slow motions. From this, that the observed glass is
thermodynamically a nonequilibrium state is concluded.

Figure 1: Glass transition represented in terms of viscosity $\eta$ (a) and
specific heat $C_{p}$ (b). Evolution of the energy barrier for atomic movement
is shown in (c).

The causes behind the above contrasting views are deeply rooted in the
fundamental problems of thermodynamics: the definition of equilibrium and TC.
The rigorous definition of equilibrium has been only recently established
Gyftopoulos and Beretta (2005), and it enables us to define TC in a consistent
manner. The author applied this principle to solids and found that the
equilibrium positions of all the atoms $\left\\{\bar{\bf R}_{j}\right\\}$ are
TCs for solids Sta . The present viewpoints are based on this quite general
conclusion Shirai (2020). The important points relevant to this study are
listed below, with contrasting the general views in glass research Ediger
(2000); Lubchenko and Wolynes (2005); Chandler and Garrahan (2010); Berthier
and Biroli (2011); Biroli and Garrahan (2013); Stillinger and Debenedetti
(2013): however, some of the views are not necessarily unique even in the
glass research.

  1. 1.

Glass is a solid: it is viewed as a frozen liquid in glass literature. An
often-claimed reason for the lack of distinction between the liquid and glass
states is that atomic positions are random in both states. However, this is
only true when the instantaneous positions of atoms ${\bf R}_{j}(t)$ are
compared. Thermodynamically, equilibrium positions $\bar{\bf R}_{j}$, which
are obtained by time averaging, have sense. They are well defined for solids,
but not for liquids.

  2. 2.

Glass is an equilibrium state: it is viewed as a nonequilibrium state in glass
literature. This is the natural consequence of the preceding conclusion.
Although the changing state in the course of time during the transition is a
nonequilibrium state, the values of TCs can be specified for each moment. When
the change in the external conditions is halted and keep it constant for a
time longer than the relaxation time, the transition state finally reaches the
equilibrium state even in the transition region of temperature.

  3. 3.

Glass transition that occurs at $T_{g}$ is a genuine phase transition: it is
viewed as an extrinsic change in glass literature. Although the transition has
dependence on the preparation conditions, this dependence, small or large, is
a common feature of all phase transitions. In molecular-beam-epitaxy method,
the growth temperature varies with the kinetics of source gases.

  4. 4.

The transition state is a mixture of liquid and solid phases: it is viewed as
a supercooled-liquid state after a sufficiently long time in glass literature.
Although the final states according to the two viewpoints are different, both
share the common thought that the transition region is a crossover region
between different phases. The term dynamic heterogeneity in glass literature
is also compatible with the present view of the mixed state.

A natural consequence of the above-described views is that the energy barrier
$E_{b}$ for atomic movement depends on the structure, that is,
$E_{b}=E_{b}(\left\\{{\bf R}_{j}\right\\})$, and hence it depends on
temperature Shi . It follows that $E_{b}$ varies strongly in the transition
region from the energy barrier of the solid state ($E_{b,g}$) to that of the
liquid state ($E_{b,l}$), as shown schematically in Fig. 1(c). The temperature
dependence of the activation energy becomes clearer in molecular-dynamic
simulations Debenedetti and Stillinger (2001); Han et al. (2020). The idea of
the $T$-dependent energy barrier was at times proposed by observing the
deviations from the Arrhenius law Ward (1937); Gutmann and Simmons (1952); Doo
; Hildebrand and Lamoreaux (1997). However, this idea did not gain wide
acceptance. The difficulty in interpreting the $T$ dependence of the
viscosity—even if the structural dependence of energy barrier is accepted—is
that the values of the activation energy at $T_{g}$ are too large on account
of the magnitude of the energies of chemical bonds, as mentioned in
Introduction. Further, the pre-exponential factor is too large: the absolute
values of $\log\eta_{0}$ are greater than 100 for organic glasses Hodge
(1994).

###  II.2 Theory of viscosity

Many theories have been proposed for explaining the viscosity of glass since
Eyring derived the quantum-mechanical formula for viscosity of liquids
Glasstone et al. (1964); Eyring (1936). These theories have been reviewed in
textbooks by Nemilov Nemilov (1995) and Rao Rao (2002). The current trend of
study has shifted to microscopic dynamics, focusing on topics such as the
mode-coupling theory, dynamic facilitation, and first-order random transition
Jackle (1986); Angell et al. (2000); Ediger (2000); Sciortino (2005);
Lubchenko and Wolynes (2005); Berthier and Biroli (2011); Stillinger and
Debenedetti (2013); Biroli and Garrahan (2013). The approach employed in this
study is the traditional one based on the reaction rate theory, with the aim
of facilitating the chemical interpretation of the viscosity of glass.

For normal liquids, formula Eq. (2), including the pre-exponential factor
$\eta_{0}$, can be deduced from the microscopic theory, without assuming any
model Glasstone et al. (1964); Eyring (1936). Eyring’s theory explains well
the viscosity of normal liquids Ewell and Eyring (1937). For an isotropic
media, the pre-exponential factor $\eta_{0}$ in Eq. (2) is expressed as

$\eta_{0}=\left(\frac{a_{0}}{\lambda}\right)^{2}\frac{h}{a_{0}^{3}}\frac{Z_{n}}{Z_{a}},$
(3)

where $a_{0}$ is the mean interatomic distance, and $\lambda$ is the average
distance between the equilibrium positions of the original and slipped states.
The product $(\lambda/a_{0})^{2}a_{0}^{3}$ may be interpreted as the
activation volume $V_{a}$, which is the volume swept by the slipped atoms.
Planck’s constant $h$ is a universal constant but is equal to $k_{\rm
B}T/\nu_{m}$ at thermodynamic equilibrium, where $\nu_{m}$ is the mean
frequency of the frequency spectrum at $T$ and is interpreted as the attempt
frequency. $Z_{n}$ and $Z_{a}$ are the partition functions of the normal and
the activated states of the slipped atoms, respectively. Although it is
difficult to calculate the partition function $Z$, the values of $Z$ in Eq.
(3) are obtained only as the ratio $Z_{n}/Z_{a}$, which is of the order of
unity in most cases, and hence, we can ignore this term. By assuming
$\lambda\approx a_{0}$, $\eta_{0}=h/V_{0}$ gives $\log\eta_{0}=-3.5$ for
$V_{0}=10\ {\rm\AA}^{3}$. This result is well in agreement with the
experimental results: in many cases, $\log\eta_{0}=-3$ to $-4$ Barrer (1943).
The activation energy $Q_{a}$ is of the order of 0.1 eV or less Ewell and
Eyring (1937); Barrer (1943).

For solids, the definition of viscosity is not so simple because of its
nonlinearity. Nevertheless, the rate theory of Eyring was widely applied to
solid dynamics beyond the original object of viscosity. Plastic deformation of
crystalline solids is explained by the rate theory, in which the motion is
mediated by dislocations Schöck (1980). For the case of metallic glass, the
motion is mediated by free volumes Spaepen (1977) or by shear transformation
zones Argon (1977). The physics involved in the processes is different from
those of liquids. Despite this, the formal expression for the rate of
deformation is almost the same as Eq. (3), if the quantities in this equation
are suitably interpreted. Hence, there is no reason not to apply this formula
for glasses in general. However, the problem in this case is that the
material-dependent “constants” are not constant during the transition, because
the structure changes during the transition.

Now, let us ask the question why the $T$ dependence of the viscosity does not
deviate from the Arrhenius law for strong glass. From the present viewpoint,
the energy barrier strongly varies with temperature, and hence a large
deviation from the Arrhenius law is expected for all glasses. The apparent
energy barrier $Q_{a}^{*}$ is obtained from the derivative of $\ln\eta$ with
respect to $1/T$,

$Q_{a}^{*}=\frac{\partial\ln\eta}{\partial(1/T)}.$ (4)

We are so accustomed to this formula to obtain energy barriers in numerous
applications that we seldom consider how the Arrhenius form is altered when
the energy barrier has $T$ dependence. Let us consider a simple case of the
linear dependence of the energy barrier on temperature,

$E_{b}(T)=E_{b,g}-b(T-T_{g,1}),$ (5)

where $b=(E_{b,g}-E_{b,l})/\Delta T_{g}$ is a constant. In the Arrhenius form,
the linear term in $T$ is canceled by the denominator in the exponent of Eq.
(2). Thus, Eq. (2) becomes

$\eta(T)=\eta_{0}e^{-b/k_{\rm B}}\exp\left(\frac{Q_{a}^{*}}{k_{\rm
B}T}\right),$ (6)

where $Q_{a}^{*}=E_{b,g}+bT_{g,1}$. Hence, $T$ dependence is not seen in
$Q_{a}^{*}$. Instead, there exists a large separation between $Q_{a}^{*}$ and
$E_{b}$. Since $\Delta T_{g}$ is small, $Q_{a}^{*}$ is approximated by

$Q_{a}^{*}\approx\frac{T_{g}}{\Delta T_{g}}(E_{b,g}-E_{b,l}).$ (7)

The factor $k=T_{g}/\Delta T_{g}$ acts as a magnification factor for the
barrier height. Although the linear term disappears from Eq. (7), its effect
appears in the pre-exponential factor as
$\eta_{0}^{\prime}=\eta_{0}e^{-b/k_{\rm B}}$. Sometimes, the entropy of atomic
migration contributes to the transport coefficients through the
$T$-independent term $e^{b/k_{\rm B}}$ in a manner similar to Eq. (6).
However, the above derivation shows that it is not appropriate to interpret
the $e^{b/k_{\rm B}}$ term in terms of entropy. In fact, $b/k_{\rm B}$ often
becomes as large as 100. It is impossible to explain this magnitude by
entropy. Even for vaporization, the increase in entropy is at most of the
order of 10 times $k_{\rm B}$. To obtain the genuine energy barrier $E_{b}$,
the term $e^{-b/k_{\rm B}}$ should be retained in the Arrhenius analysis.
Thus, $E_{b}$ can be obtained by

$E_{b}(T)=k_{\rm B}T\ln\left(\frac{\eta(T)}{\eta_{0}}\right),$ (8)

provided that $\eta_{0}$ is independent of $T$. Unfortunately, $\eta_{0}$ may
not be expected to remain constant during phase transition because the
structure changes and does the size $\lambda$ of the moving unit. In addition,
the ratio $Z_{n}/Z_{a}$ may not be ignored because the moving units become
more collective motions for solids. This problem has to be solved.

##  III Width of glass transition

The present theory predicts that the apparent activation energy $Q_{a}^{*}$ is
magnified by the factor $k=T_{g}/\Delta T_{g}$. Let us examine this
magnification in $Q_{a}^{*}$ by analyzing the experimental data. The data for
this purpose are taken from measurements of the specific heat. While numerous
data have been accumulated for $T_{g}$, data for $\Delta T_{g}$ are very rare.
The width is sensitive to the shape of the $C$-$T$ curve in the transition
region, which is largely affected by the conditions of sample preparation and
measurement Moynihan et al. (1976). It is, therefore, important to use the
references in which the experimental conditions are well documented. Moynihan
compiled data of $\Delta T_{g}$ for 17 inorganic glasses with $T_{g}$ well
above room temperature Moynihan (1995). These glasses include chalcogenides,
heavy-metal fluorides, and network oxide glasses. He took special care when
collecting experimental data: i.e., only the data obtained under common
conditions, such as within an acceptable range of heating rate, were
considered. He found that there is a good correlation between $Q_{a}^{*}$ in
viscosity measurement and $\Delta T_{g}$ in heat-capacity measurement

$\frac{Q_{a}^{*}}{k_{\rm B}}\Delta\left(\frac{1}{T_{g}}\right)=4.8\ \pm 0.4.$
(9)

The values of $Q_{a}^{*}$ are obtained in the range $11\leq\log\eta(T_{g})\leq
12$, where the $T$ dependence of viscosity obeys the Arrhenius law. The
relationship between $T_{g}/\Delta T_{g}$ and $Q_{a}^{*}/k_{\rm B}T_{g}$ which
is taken from Moynihan’s data is plotted in Fig. 2. The acronyms and the
chemical compositions are explained in his paper. There is a good correlation
between $T_{g}/\Delta T_{g}$ and $Q_{a}^{*}/k_{\rm B}T_{g}$.

Figure 2: Plot of Moynihan’s data showing the relationship between
$T_{g}/\Delta T_{g}$ and $Q_{a}^{*}/k_{\rm B}T_{g}$ Moynihan (1995). The
sources of the original data are quoted in Ref. Moynihan (1995).

From the approximation of Eq. (7), Eq. (9) is rewritten as

$\frac{E_{b,g}-E_{b,l}}{k_{\rm B}T_{g}}=4.8.$ (10)

This ratio seems to be reasonable when compared with the range of energy
barrier $E_{b}$ of impurity diffusion. In silica glass, the energy barriers
$E_{d}$ of impurity diffusions are reported to range from 0.3 to 0.8 eV Barrer
(1934). Since for silica glass $T_{g}=1450$ K, the ratio $E_{d}/k_{\rm
B}T_{g}$ ranges from 2 to 6. Although diffusion and viscose motions are
different modes of motions, the energy barriers should not be so different
Avramov et al. (2009); Brillo et al. (2011). Thus, the relationship shown in
Eq. (10) must give reasonable values for $E_{b}$. This infers that the
previous values for the activation energy for glass transition were
significant overestimations.

Among various methods available, one definition of fragility is
$m=(\partial\ln\eta/\partial(1/T))/k_{\rm B}T_{g}$, meaning the activation
energy normalized with the glass-transition temperature Böhmer et al. (1993).
Hence the relationship Eq. (9) is also rewritten as follows:

$m=4.8k.$ (11)

Although the factor was determined as 4.8 from a certain group of glasses,
this may not largely change among various glasses: Ito et al. showed that this
is indeed the case Ito et al. (1999). Thus, one can immediately see that
fragility represents the magnification factor $k$. Therefore, the greater
fragility, the more magnified $Q_{a}^{*}$ is compared to the true value
$E_{b}$. Therefore, fragile glasses exhibit large $Q_{a}^{*}$, in spite of
their low melting temperatures. This naturally leads to the following
interpretation of fragility $m$: that is, it represents how rapidly the energy
barrier varies with varying $T$. This interpretation is more appealing because
the energy barrier is the standard terminology in solid state physics and
chemistry. A similar interpretation for the $m$ was inferred by Dyre and Olsen
Dyre and Olsen (2004). The present study gives an additional meaning of $m$ as
the magnification factor for the activation energy Researchers attempted to
find correlations between fragility with other properties of glasses. Fujimori
and Oguni found the correlation of $m$ with a special index representing the
difference between $\alpha$ and $\beta$ relaxations Fujimori and Oguni (1995).
Scopigno at al. found another correlation of $m$ with the decorrelation of the
density fluctuations Scopigno et al. (2003). These correlations can be
understood better by using a familiar term, namely, energy barrier.

##  IV Analysis of the temperature dependence of viscosity

###  IV.1 Silicate glass

Let us examine the full-scale temperature dependence of viscosity for three
classes of glasses. The first class of glasses comprises silicate glasses,
which are typically strong glasses. It is said that the $T$ dependence of
$\eta$ for strong glasses obeys the Arrhenius law, but it is true only in a
relative sense compared with fragile glasses. If the $T$ dependence of $\eta$
is examined over a wide range of $T$, one sees a large variation in the
activation energy: for example, the $Q_{a}^{*}$ of silica glass increases from
4.0 eV above $T=2000^{\circ}$C to 8.1 eV at $T=1400^{\circ}$C as temperature
declines Mackenzie (1961); Hetherington et al. (1964); Fontana and Plummer
(1968); Laughlin and Uhlmann (1972).

Figure 3: Apparent activation energy $Q_{a}^{*}$ of silicate glasses obtained
by the conventional method. Data are for soda-lime-silicate (SLS) glass
samples, except for Kawamura’s data. Kawamura’s samples are high-level
radioactive waste (HLW) glass. The sources of data are as follows: Lillie
Lillie (1931), Jones Jones (1944), Napolitano Napolitano and Hawkins (1964),
Mazurin Mazurin and Potselueva (1975), and Kawamura Kawamura et al. (2014).

Figure 3 shows the apparent activation energies $Q_{a}^{*}$ obtained using Eq.
(4) for soda-lime-silicate (SLS) glass, as reported by several authors. The
chemical composition of SLS glass is formally ${\rm
SiO_{2}:Na_{2}O:CaO=75:15:10}$, and the transition temperature $T_{g}$ is
$530^{\circ}$C. The numerical data of $\eta(T)$ were retrieved from the
original figures by using a digitizer. The retrieved data were then processed
by fitting them with smoothing functions, and $Q_{a}^{*}$ was obtained by
taking the derivatives of the fitting function according to Eq. (4). All the
original data and the subsequent process of analysis are provided in
Supplemental material. As seen in Fig. 3, the apparent activation energy
$Q_{a}^{*}$ increases as temperature decreases to $T_{g}$. As the temperature
approaches $T_{g}$, $Q_{a}^{*}$ reaches about 6.8 eV and then quickly
decreases. The maximum value is more than twice the value at
$T=1200^{\circ}$C. The linearity in the Arrhenius plot holds only for the
range of $11\leq\log\eta\leq 14$, which corresponds to a temperature range of
$600^{\circ}$C $\geq T\geq 500^{\circ}$C.

Figure 4: Energy barrier $E_{b}$ of SLS glass. The same symbols as those used
in Fig. 3 are also used here. All data are calculated by assuming
$\log\eta_{0}=0$. Under the assumption that $\log\eta_{0}=10.4$, the corrected
values for Kawamura’s data are reduced to the green line.

Let us calculate the corrected value $E_{b}$ according to Eq. (8). The
constant $\eta_{0}$ is not known. The lowest value of the measured viscosity
cited above is about $\log\eta=2$, which was reported by Lullie Lillie (1931).
The pre-exponential factor $\eta_{0}$ is less than this value but is larger
than a typical value of normal liquids, i.e., $-3.5$. Since silicate glasses
are among the most viscous materials, tentatively, $\log\eta_{0}=0$ was used.
The calculated values are plotted in Fig. 4. The maximum value of $E_{b}$
reduces significantly to 2.2 eV from the corresponding value $Q_{a}^{*}$ of
6.8 eV. This $E_{b}$ value of 2.2 eV is already smaller than the Si-O bond
energy of SiO2 (4.6 eV). If an increase in $\eta_{0}$ with decreasing $T$ is
taken into account, the value $E_{b}$ is further reduced. The $E_{b}$ value
can also be estimated by using Eq. (7). For the transition width $\Delta
T_{g}$ of the SLS glass, the value of NBS710 ($T_{g}/\Delta T_{g}=17$, as
shown in Fig. 2) is adopted. Equation (7) leads to $E_{b,g}-E_{b,l}=0.37$ eV.
The model shown in Eq. (5) assumes that the change in $E_{b}$ occurs only
within $\Delta T_{g}$. However, as seen in Fig. 4, the change in $E_{b}$
begins from far higher temperatures and is more gradual against $T$. In this
respect, Eq. (7) may be an overcorrection. The true value $E_{b}$ lies between
0.37 and 2.2, which covers the diffusion barrier $E_{d}$ of silica glass
($0.3\sim 0.8$ eV) mentioned above.

A notable observation in Fig. 4 is the temperature dependence of $E_{b}$ below
$T_{g}$. It is not easy to measure viscosity values larger than $\log\eta=13$.
Despite this difficulty, viscosity below $T_{g}$ were measured even in early
studies Stanworth (1937); Jones (1944, 1949). For SLS glass, Jones reported a
significant reduction in $Q_{a}^{*}$ from 2.3 eV above $T_{g}$ to 0.25 eV at
$T=350^{\circ}$C, at which $\eta$ increases to about $\log\eta=18$ Jones
(1944, 1949). The values reported by Jones are shown in Fig. 3: these values
are obtained by the present analysis employing a smoothing functions, and
hence the values are slightly different from the values reported by Jones. A
similar reduction was reported by Shen et al Shen et al. (2003): a reduction
from $Q_{a}^{*}=$5.2 eV above $T_{g}$ to 1.2 eV at $T=450^{\circ}$C was
observed for SLS glass. Surprisingly, on applying the correct form of Eq. (8),
these quick reductions in $Q_{a}^{*}$ below $T_{g}$ are drastically modified
so that they are nearly constant, as shown in Fig. 4. This is consistent with
the present model shown in Fig. 1(c). This behavior is reasonable because the
structure does not change once the glass substance is frozen. There are some
literatures describing the activation energy as being saturated around $T_{g}$
Mazurin et al. (1982); Nemilov (1995), which means a recovery of the Arrhenius
law below $T_{g}$. Recently, Kawamura et al. performed viscosity measurements
at temperatures as low as $270^{\circ}$C, which may be the lowest temperature
ever reported for silicate glasses Kawamura et al. (2014). They employed the
fiber-bending method, which was developed by Koide Koide et al. (1994). They
performed measurements for a simulated high-level radioactive waste (HLW)
glass. The main difference between HLW and SLS glass is inclusion of B2O3 in
HLW. Hence, it is reasonable to observe that the viscosity of HLW glass was
lower than that of SLS glass. If, for Kawamura’s data, $\log\eta_{0}$ is
calculated to make $\eta(T)$ constant below $T_{g}$, the pre-exponential
factor is obtained as $\log\eta_{0}=10.4$, as indicated by the green line in
Fig. 4.

All the presented data were measured by independent authors and by different
methods. Therefore, the agreement between the values of $E_{b}$ below as well
as above $T_{g}$ validates the reliability of the obtained $T$ dependence. The
energy barrier $E_{b}$ becomes saturated below $T_{g}$. By taking all the
above results together, it is likely that $E_{b}$ of SLS glass is no more than
1 eV. Furthermore, the present analysis shows that no divergence occurs at
$T_{0}$. This conclusion agrees with the conclusions of recent studies on the
glass transition Hecksher et al. (2008); Zhao et al. (2013); Ponga et al.
(2015).

###  IV.2 Organic glass

The next class of glasses to be examined comprises organic glasses. Generally,
the glasses belonging to this class have low-melting temperatures that are
near or lower than room temperature. Most of them are fragile glasses, and
hence the apparent activation energy $Q_{a}^{*}$ strongly depends on $T$. When
$T$ approaches to $T_{g}$ from high temperatures, $Q_{a}^{*}$ increases
quickly, as shown in the Angell plot. Large values of $Q_{a}^{*}$ are reported
at $T_{g}$: e.g., $Q_{a}^{*}$ = 5.5 eV for glucose and 1.1 eV for glycerol
Davies and Jones (1953a). More examples are provided in the review by Hodge,
who has presented many values of $Q_{a}^{*}$ in different ways depending on
the definition of the activation energy: apparent activation energies
exceeding 10 eV are listed Hodge (1994). In this work, $o$-terphenyl is
examined as an example of organic glasses. This is because the viscosity of
this material was measured in a wide range from $\log\eta=-0.5$ to $13$, which
covers the whole range of the Angell plot. Glass transition occurs at
$T_{g}=239$K with the width $\Delta T_{g}=5$K. For model calculations of the
$T$ dependence of $o$-terphenyl, see Ref. Nemilov (1995). The inset of Fig. 5
shows $Q_{a}^{*}$ of $o$-terphenyl as obtained by the conventional method
using Eq. (4).

Figure 5: Energy barrier $E_{b}$ of $o$-terphenyl glass. $T_{g}=239$K. The
inset shows the apparent activation energy $Q_{a}^{*}$ obtained using the
traditional Arrhenius plot. The data are taken from Ref. Nemilov (1995).

The data $\eta(T)$ were retrieved from Nemilov’s textbook (Fig. 44 of Ref.
Nemilov (1995)): the plot data actually consist of three sets of data provided
by Laughlin Laughlin and Uhlmann (1972), Greet Greet and Turnbull (1967), and
Cukierman Cukierman et al. (1973). The apparent activation energy $Q_{a}^{*}$
increases up to 4 eV, which is close to that of silicate glass. In the
literature, this large $Q_{a}^{*}$ for organic glasses is at times explained
by the collective modes of a large moving units. This explanation is not
convincing. A collective mode can alter $\lambda$ in the pre-exponential
factor in Eq. (3). However, each atom constituting the moving unit receives
equally thermal energy $k_{\rm B}T$. Hence, it is unlikely that the activation
energy increases with the increase in the size of moving units. This
unreasonably large value can, however, be corrected by the present theory. By
applying Eq. (8), a significant reduction from $Q_{a}^{*}$ is obtained. This
is shown in Fig. 5. In this case, $\log\eta_{0}=-2$ is used. The maximum
$E_{b}$ is 0.8 eV, which is within a reasonable range of the energy barrier.
If the multiplication factor, in this case $k=50$, is adopted, the value
$E_{b}$ becomes as small as 0.1 eV, when $E_{b,l}$ is ignored. Now, it is
clear that the large values of the apparent activation energy $Q_{a}^{*}$ that
are common for organic glasses can be ascribed to the narrow width $\Delta
T_{g}$.

###  IV.3 Metallic glass

The third example comprises metallic glasses. Metallic glasses have fragility
intermediate between that of strong covalent glasses and fragile organic
glasses Busch et al. (2007). The apparent activation energy $Q_{a}^{*}$ of the
glass transition of metallic glasses ranges from 2 to 8 eV near $T_{g}$
Johnson et al. (2007). The plastic deformation of crystalline materials is
mediated by dislocations, whereas for metallic glasses it is madiated by the
so-called shear transformation zone (STZ) Argon (1977). The approach for
describing the $T$ dependence of the viscosity is opposite to that employed in
Sec. II.2, where the starting state is the liquid state. For the area of
metallic glasses, the study starts from the solid state Schuh et al. (2007).
The formula for viscosity is formally the same as that given in Eq. (2). The
activation volume $V_{a}$ is considered to be that of an STZ. For ZrTiCuNi
glass, a typical STZ includes 20–30 atoms Schuh et al. (2007).

The temperature dependence of the viscosity has been gleaned from recent
experiments on metallic glasses. A theoretical model was proposed by Johnson
et al., who considered a formula for the energy barrier based on the elastic
moduli of the solid state and extended it to the liquid state Demetrios et al.
(2006); Johnson et al. (2007). They adapted the exponential decay for the $T$
dependence of the energy barrier, obtaining a formula

$\frac{\eta}{\eta_{0}}=\exp\left\\{\frac{W_{g}}{k_{\rm
B}T}\exp\left[2n\left(1-\frac{T}{T_{g}}\right)\right]\right\\},$ (12)

for $T>T_{g}$. Here, $W_{g}$ is the energy barrier at $T=T_{g}$, and $n$ is a
fitting parameter whose value is the order of unity Johnson et al. (2007).
When $T$ approaches $T_{g}$, the term $\exp\left[2n(1-T/Tg)\right]$ in Eq.
(12) can be expanded with respect to $\Delta T=T-T_{g}$, resulting in

$\frac{\eta}{\eta_{0}}=\exp\left\\{\frac{W_{g}}{k_{\rm
B}T}\left(1+2n-2n\frac{T}{T_{g}}\right)\right\\}.$ (13)

This has the same form as the linear approximation of Eq. (5) with a
magnification factor $k=1+2n$. Numeric examples are given by them: for ${\rm
Pd_{77.5}Cu_{6}Si_{16.5}}$, $T_{g}=634$K, $\log\eta_{0}=-2.11$, and $n=1.67$.
This leads to the magnification factor $k$ of $4.3$. Hence, the correction for
$E_{b}$ is not too large compared with those of other classes of glasses.

An interesting point regarding metallic glasses is that viscosity below
$T_{g}$ is sometimes measured because of the practical interest in the creep
phenomenon. Taub and Spaepen found that the activation energy $Q_{a}^{*}$ of
viscosity of ${\rm Pd_{82}Si_{18}}$ is almost constant at about 2.0 eV in a
temperature range from 420 to 540 K Taub and Spaepen (1980); ${\rm
Pd_{82}Si_{18}}$ is a metallic glass with $T_{g}=634$ K Wang et al. (2017).
Interestingly, the value $Q_{a}^{*}$ does not change by annealing at various
temperatures. This is consistent with the present assumption, Eq. (5). There
is no sign to exhibit the divergent behavior of the VFT law in this case too.

##  V Discussion

Although phase transition was not the original focus of this study, the
findings of this study have further implication to this issue. The analyses in
Sec. IV established that none of the experimental data on viscosity below
$T_{g}$ showed any sign of exponential divergence at a finite temperature. The
$T$ dependence of $\eta(T)$ seems to obey the Arrhenius law rather well.
Indirect evidence for this non divergence was recently reported Hecksher et
al. (2008); Zhao et al. (2013); Ponga et al. (2015).

Both $T_{0}$ and $T_{K}$ are obtained by extrapolating viscosity and entropy,
respectively, measured at $T>T_{g}$. This extrapolation loses its validity
when one accepts the experimental transition occurring at $T_{g}$ as a genuine
phase transition. Nobody considers that an extrapolation of the $T$ dependence
of entropy ($S\sim\ln T$) of an ideal gas to $T=0$ is meaningful: such
extrapolation diverges negatively. The repeated assertion is that the observed
glass transition occurring at $T_{g}$ is not an intrinsic property of a glass
because $T_{g}$ is affected by the preparation conditions. This subject was
already discussed in Ref. Shirai (2020). The essential points are repeated
herein. First, the kinetic nature of the transition is common to all
liquid/solid phase transitions. Crystallization is determined by the
competition of the kinetic effect of entropy and the potential effect on
constraining atomic motions. Second, preparation dependence is common in every
transition, though the effect is outstanding for glasses. Freezing of water to
ice is a well-known example. The graphite-diamond transition temperature is
largely affected by its kinetics. Any crystallization accompanies a kinetic
process of nucleation that can be altered by external conditions. The
reference temperature $T_{0}$ in the VFT formula is only a fitting parameter
to reproduce the temperature dependence of viscosity above $T_{g}$.

##  VI Conclusion

In usual phase transitions, the energy barrier for atomic movement changes
abruptly. In fact, the transition occurs at a fixed temperature, such as
$T_{m}$, which hides this change from observation. In contrast, for glass
transition, the continuous structural change occurs in a certain range of
temperature, i.e., $\Delta T_{g}$. This structural change causes a change in
the energy barrier $E_{b}$ in a continuous manner, and this causes $\eta(T)$
to deviate from the Arrhenius law. The present theory predicts that the
observed value of the activation energy $Q_{a}^{*}$ largely overestimates the
energy barrier $E_{b}$. This prediction has been validated by examining
available experimental data. The degree of deviation depends mainly on the
magnification factor $k=T_{g}/\Delta T_{g}$. Generally, fragile glasses
exhibit sharp transitions with narrow widths $\Delta T_{g}$. This explains the
extraordinarily large values for the observed $Q_{a}^{*}$ of fragile glasses:
e.g. the magnification factor is on the order of magnitude 50. This
magnification also occurs in strong glasses, whereas the magnification factor
is not very large. In either case, the genuine energy barrier $E_{b}$ is
estimated to be less than 1 eV, which lies in a reasonable range of the
chemical energy of liquids and solids. The analysis has provided another
interpretation for fragility of glass: fragility indicates how fast the energy
barrier varies with varying temperature. Thus, the present result enables us
to interpret the behavior of the viscosity of glass on the same ground of
energetic approach that is commonly used in solid-state physics and chemistry.
The results also provide a starting point of future study by first-principles
calculations.

A shortcoming of the current status of this theory is lack of the concrete
formula for the pre-exponential factor $\eta_{0}$. Since the present
assumption of Eq. (5)—the energy barrier changes only within $T_{g}$—is not
very good, the lack of the concrete formula for $\eta_{0}$ can cause a serious
problem. Experimental data given in Sec. IV show that the energy barrier
begins to change at temperatures far higher than $T_{g}$. Clearly, this
shortcoming should be solved.

###### Acknowledgements.

This work was performed under the Research Program of “Dynamic Alliance for
Open Innovation Bridging Human, Environment and Materials” in “Network Joint
Research Center for Materials and Devices”. The author also thanks Enago
(www.enago.jp) for the English language review.

## References

  * Angell et al. (2000) C. A. Angell, K. L. Ngai, G. B. McKenna, P. F. McMillan, and S. W. Martin, Appl. Phys. Rev. 88, 3113 (2000).
  * Lubchenko and Wolynes (2005) V. Lubchenko and P. G. Wolynes, Annu. Rev. Phys. Chem. 58, 235 (2005).
  * Berthier and Biroli (2011) L. Berthier and G. Biroli, Rev. Mod. Phys. 83, 587 (2011).
  * Stillinger and Debenedetti (2013) F. H. Stillinger and P. G. Debenedetti, Annu. Rev. Condens. Matter Phys. 4, 263 (2013).
  * Ediger (2017) M. D. Ediger, J. Chem. Phys. 147, 210901 (2017).
  * Niss and Hecksher (2018) K. Niss and T. Hecksher, J. Chem. Phys. 149, 230901 (2018).
  * Royall et al. (2018) C. P. Royall, F. Turci, S. Tatsumi, J. Russo, and J. Robinson, J. Phys.: Condens. Matter 30, 363001 (2018).
  * Vogel (1921) H. Vogel, Physikalische Zeitscrift 22, 645 (1921).
  * Fulcher (1925) G. S. Fulcher, J. Am. Ceram. Soc. 8, 339 (1925).
  * Tammann and Hesse (1926) G. Tammann and W. Z. Hesse, Z. Anor. Allg. Chem. 156, 245 (1926).
  * Angell (1997) C. A. Angell, J. Res. Natl. Stand. Tech. 109, 171 (1997).
  * Ngai and Yamamuro (1999) K. L. Ngai and O. Yamamuro, J. Phys. Chem. 111, 10403 (1999).
  * Ito et al. (1999) K. Ito, C. T. Moynihan, and C. A. Angell, Nature 398, 492 (1999).
  * Martinez and Angell (2001) L.-M. Martinez and C. A. Angell, Nature 410, 663 (2001).
  * Sastry (2001) S. Sastry, Nature 409, 164 (2001).
  * Tanaka (2003) H. Tanaka, Phys. Rev. Lett. 90, 055701 (2003).
  * Wang et al. (2006) L.-M. Wang, C. A. Angell, and R. Richert, J. Chem. Phys. 125, 074505 (2006).
  * Hecksher et al. (2008) T. Hecksher, A. I. Nielsen, N. B. Olsen, and J. C. Dyre, Nature Phys. 4, 737 (2008).
  * Glasstone et al. (1964) S. Glasstone, K. J. Laider, and H. Eyring, _The Theory of Rate Processes: The kinetics and chemical reactions, viscosity, diffusion and electrochemical phenomena_ (McGraw-Hill, New York, 1964).
  * Ewell and Eyring (1937) R. H. Ewell and H. Eyring, J. Chem. Phys. 5, 726 (1937).
  * Davies and Jones (1953a) R. O. Davies and G. O. Jones, Proc. Roy. Soc. A 217, 26 (1953a).
  * Davies and Jones (1953b) R. O. Davies and G. O. Jones, Adv. Phys. 2, 370 (1953b).
  * Hodge (1994) I. M. Hodge, J. Non-Cryst. Solids 169, 211 (1994).
  * Nemilov (1995) S. V. Nemilov, _Thermodynamic and Kinetic Aspects of the Vitreous State_ (CRC Press, Boca Raton, 1995).
  * Laughlin and Uhlmann (1972) W. T. Laughlin and D. R. Uhlmann, J. Phys. Chem. 76, 2317 (1972).
  * Angell and Sichina (1976) C. A. Angell and W. Sichina, Ann. N.Y. Acad. Sci. 279, 53 (1976).
  * Angell (1988) C. A. Angell, J. Phys. Chem. Solids 49, 863 (1988).
  * (28) Yet another method of normalization is proposed with respect to $T_{m}$: Z. Nussinov, Phil. Mag. 97, 1509 (2017).
  * Bridgman (1950) P. W. Bridgman, Rev. Mod. Phys. 22, 56 (1950).
  * Kestine and Rice (1970) J. Kestine and J. R. Rice, in _A Critical Review of Thermodynamics_ , edited by E. B. Stuart, B. Gal-Or, and A. J. Brainard (Mono Book Corp., Baltimore, 1970), pp. 275–298.
  * Maugin and Muschik (1994) G. A. Maugin and W. Muschik, J. Non-Equilib. Thermodyn. 19, 217 (1994), 19, 250 (1994).
  * (32) K. Shirai, cond-mat.stat-mech/1812.08977.
  * Shirai (2020) K. Shirai, J. Phys. Commun. 4, 085015 (2020).
  * (34) K. Shirai, cond-mat.stat-mech/2003.13873.
  * Kauzmann (1948) W. Kauzmann, Chem. Rev. 43, 219 (1948).
  * Angell (1995) C. A. Angell, Science 267, 1924 (1995).
  * Mazurin (2007) O. V. Mazurin, Glass Phys. Chem. 33, 22 (2007).
  * Moynihan et al. (1976) C. T. Moynihan, A. J. Easteal, M. A. DeBolt, and J. Tucker, J. Am. Ceram. Soc. 59, 12 (1976).
  * Gyftopoulos and Beretta (2005) E. P. Gyftopoulos and G. P. Beretta, _Thermodynamics - Foundations and Applications_ (Dover Pub., New York, 2005).
  * Ediger (2000) M. D. Ediger, Annu. Rev. Chem. Chem. 51, 99 (2000).
  * Chandler and Garrahan (2010) D. Chandler and J. P. Garrahan, Annu. Rev. Phys. Chem. 61, 191 (2010).
  * Biroli and Garrahan (2013) G. Biroli and P. Garrahan, J. Chem. Phys. 138, 12A301 (2013).
  * Debenedetti and Stillinger (2001) P. G. Debenedetti and F. H. Stillinger, Nature 410, 259 (2001).
  * Han et al. (2020) D. Han, D. Wei, J. Yang, H.-L. Li, M.-Q. Jiang, Y.-J. Wang, L.-H. Dai, and A. Zaccone, Phys. Rev. B 101, 014113 (2020).
  * Ward (1937) A. G. Ward, Trans. Faraday Soc. 33, 88 (1937).
  * Gutmann and Simmons (1952) F. Gutmann and L. M. Simmons, J. Appl. Phys. 23, 977 (1952).
  * (47) A. K. Doolittle, J. Appl. Phys. 22, 1471 (1951); A. K. Doolittle and D. B. Doolittle, J. Appl. Phys. 28, 901 (1957).
  * Hildebrand and Lamoreaux (1997) J. H. Hildebrand and R. H. Lamoreaux, Proc. Nat. Acad. Sci. USA 73, 988 (1997).
  * Eyring (1936) H. Eyring, J. Chem. Phys. 4, 283 (1936).
  * Rao (2002) K. J. Rao, _Structural Chemistry of Glasses_ (Elsevier, Amsterdam, 2002).
  * Jackle (1986) J. Jackle, Rep. Prog. Phys. 49, 171 (1986).
  * Sciortino (2005) F. Sciortino, J. Stat. Mech. 2005, P05015 (2005).
  * Barrer (1943) R. M. Barrer, Trans. Faraday Soc. 39, 48 (1943).
  * Schöck (1980) G. Schöck, in _Dislocations in Solids_ , edited by F. R. N. Nabarro (Elsevier, Amsterdam, 1980), p. 63.
  * Spaepen (1977) F. Spaepen, Acta Matell. 25, 407 (1977).
  * Argon (1977) A. S. Argon, Acta Matell. 27, 47 (1977).
  * Moynihan (1995) C. T. Moynihan, J. Am. Ceram. Soc. 76, 1081 (1995).
  * Barrer (1934) R. M. Barrer, J. Chem. Soc. p. 378 (1934).
  * Avramov et al. (2009) I. Avramov, T. Vassilev, and I. Penkov, J. Non-Cryst. Solids 355, 10 (2009).
  * Brillo et al. (2011) J. Brillo, A. I. Pommrich, and A. Meyer, Phys. Rev. Lett. 107, 165902 (2011).
  * Böhmer et al. (1993) R. Böhmer, K. L. Ngai, C. A. Angell, and D. J. Plazek, J. Chem. Phys. 99, 4201 (1993).
  * Dyre and Olsen (2004) J. C. Dyre and N. B. Olsen, Phys. Rev. E 69, 042501 (2004).
  * Fujimori and Oguni (1995) H. Fujimori and M. Oguni, Solid State Commun. 94, 157 (1995).
  * Scopigno et al. (2003) T. Scopigno, G. Ruocco, F. Sette, and G. Monaco, Science 302, 849 (2003).
  * Mackenzie (1961) J. D. Mackenzie, J. Am. Ceram. Soc. 44, 598 (1961).
  * Hetherington et al. (1964) G. Hetherington, K. H. Jack, and J. Kennedy, Phys. Chem. Glasses 5, 130 (1964).
  * Fontana and Plummer (1968) E. H. Fontana and W. A. Plummer, Phys. Chem. Glasses 7, 139 (1968).
  * Lillie (1931) H. R. Lillie, J. Am. Ceram. Soc. 14, 502 (1931).
  * Jones (1944) G. O. Jones, J. Soc. Glass Tech. 28, 432 (1944).
  * Napolitano and Hawkins (1964) A. Napolitano and E. G. Hawkins, J. Res. Natl. Bur. Stand. 68A, 439 (1964).
  * Mazurin and Potselueva (1975) O. V. Mazurin and L. N. Potselueva, Sov. J. Glass Phys. Chem. 1, 494 (1975).
  * Kawamura et al. (2014) K. Kawamura, M. Koide, and K. Matsusita, Nihon-Genshiryoku-Gakkai Shi (J. At. Energy Soc. Jpn.) 13, 27 (2014), in Japanese.
  * Stanworth (1937) J. E. Stanworth, J. Soc. Glass Tech. 21, 442 (1937).
  * Jones (1949) G. O. Jones, Rep. Prog. Phys. 12, 133 (1949).
  * Shen et al. (2003) J. Shen, D. J. Green, R. E. Tressler, and D. L. Shellean, J. Non-Cryst. Solids 324, 277 (2003).
  * Mazurin et al. (1982) O. V. Mazurin, Y. K. Startsev, and S. V. Stoljar, J. Non-Cryst. Solids 52, 105 (1982).
  * Koide et al. (1994) M. Koide, R. Sato, T. Komatsu, and K. Matusita, J. Non-Cryst. Solids 177, 427 (1994).
  * Zhao et al. (2013) J. Zhao, S. L. Simon, and G. B. McKenna, Nat. Commun. 4, 1783 (2013).
  * Ponga et al. (2015) E. A. Ponga, C. Rodriguez-Tinoco, G. Cerullo, C. Ferrante, J. Roriguez-Viejo, and T. Scopigno, Proc. Natl. Acad. Sci. 112, 2331 (2015).
  * Greet and Turnbull (1967) R. J. Greet and D. Turnbull, J. Chem. Phys. 46, 1243 (1967).
  * Cukierman et al. (1973) M. Cukierman, J. W. Lane, and D. R. Uhlmann, J. Chem. Phys. 59, 3639 (1973).
  * Busch et al. (2007) R. Busch, J. Schroers, and W. H. Wang, MRS Bull. 32, 620 (2007).
  * Johnson et al. (2007) W. L. Johnson, M. D. Demetriou, J. S. Harmon, M. L. Lind, and K. Samwer, MRS Bull. 32, 644 (2007).
  * Schuh et al. (2007) C. A. Schuh, T. C. Hufnagel, and U. Ramamurty, Acta Materialia 55, 4067 (2007).
  * Demetrios et al. (2006) M. D. Demetrios, J. S. Harmon, M. Tao, G. Dutan, K. Samwer, and W. L. Johnson, Phys. Rev. Lett. 97, 065502 (2006).
  * Taub and Spaepen (1980) A. I. Taub and F. Spaepen, Acta Matell. 28, 1781 (1980).
  * Wang et al. (2017) X. Wang, M. Zeng, N. Nollmann, G. Wilde, Z. Tian, and C. Tang, AIP Adv. 7, 095108 (2017).