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**Heterophase liquid states: Thermodynamics,** **structure, dynamics** ^{∗}

### A.S. Bakai

NSC Kharkiv Institute of Physics and Technology, 61108 Kharkiv, Ukraine Received December 4, 2013, in ﬁnal form June 17, 2014

An overview of theoretical results and experimental data on the thermodynamics, structure and dynamics of the heterophase glass-forming liquids is presented. The theoretical approach is based on the mesoscopic het- erophaseﬂuctuations model (HPFM) developed within the framework of the bounded partition function ap- proach. The Fischer cluster phenomenon, glass transition, liquid-liquid transformations, parametric phase dia- gram, cooperative dynamics and fragility of the glass-forming liquids is considered.

**Key words:**glass-forming liquids, glass transition, Fischer’s cluster, polymorphism, parametric phase diagram
**PACS:**78.35.+c, 64.70.Pf

**1. Introduction**

Structure of a glass-forming liquid and glass possesses a short-range and medium-range order (SRO
and MRO) rather than a long-range order (LRO). Below the crystallization temperature,*T**m*, precautions
have to be taken to avoid crystallization or a quasi-crystalline structure formation and to prevent the
supercooled liquid state down to the glass transition. Therefore, a liquid can be transformed into amor-
phous (glassy) solid only if cooling is fast enough to avoid crystallization. As a result, the liquid is non-
equilibrium and unstable at the glass transition. For this reason a description of the glass transition can-
not be based on the canonic Gibbs statistics. A palliative approach based on the bounded statistics can be
formulated as follows.

If the cooling time is much longer than the equilibration time of the liquid structure on scale*ξ*[let us
denote this time by*τ*(*ξ*)] and no signiﬁcant structural correlation occurs on scales*r*>*ξ*, one can consider
the glass transition as a sequence of transformations of the structure states which are equilibrated just on
the scales*r*<*ξ*. Statistical description of such a liquid can be developed if we exclude from the statistics
the states with the correlation scale*r*>*ξ*and, on the other hand, ensure that the observation time,*τ*obs, is
much longer than*τ*(*ξ*). In this case, the Gibbs partition function can be replaced by the bounded partition
function which is used then to determine the free energy of the partially equilibrated liquid. Limitation of
the phase space due to the exclusion of the states with correlation lengths*r*>*ξ*leads to an increase of the
free energy of the equilibrium state. The standard Gibbs statistics restores with*ξ*→ ∞. The observation
time limits from above the scale of the relaxation time*τ*(*ξ*)and, consequently, the scale*ξ*, because*τ*(*ξ*)
increases∼*ξ*^{θ}(the exponent*θ*depends on the features of the relaxation kinetics).

The spatial scale of the SRO,*ξ*SRO, is minimal among the possible correlation lengths in the liquid. Ac-
cordingly,*τ*(*ξ*SRO)is the shortest structure relaxation time because it is controlled by rearrangement of a
comparatively small number of directly interacting molecules. The formation of longer correlations, with
*ξ*≫*ξ*SRO, which involves a large number of molecules in rearrangement and is driven by relatively weak
multi-molecular forces, takes much longer than*τ*(*ξ*SRO)time. The liquid or glass, which is equilibrated on
the scale*r*∼*ξ*SROwithout considerable correlations on larger scales, is the minimally ordered amorphous
state which can be considered using the bounded statistics method. For this reason, as the ﬁrst step, the
bounded partition function should be considered taking into account the states with equilibrated SRO.

∗Dedicated to Prof. C.A. Angell on the occasion of his80^{th}birthday.

It is experimentally established that the glass-forming liquids are heterophase (their structure con-
sists of the mutually transforming ﬂuid-like and solid-like substructures). Observations of the het-
erophase structure of glass-forming liquids are numerous. Among signiﬁcant observations of the last
decades we should mention the formation of the Fischer cluster (fractal aggregate of the solid-like HPF
in glass-forming organic liquids and polymers [1–10]), evolutive HPF in supercooled triphenyl phosphit
observed in [11], and others^{1}.

Many types of the SRO usually coexist in glasses. In Bernal’s mechanical model of the dense random
packing of hard spheres, six types of the local order (Bernal’s holes) are statistically signiﬁcant and nearly
one third of them are non-crystalline [13]. Similar results are obtained using computer simulations of
liquids and glasses with different interatomic potentials [14–18]^{2}.

A wide spectrum of relaxation times in glass-forming liquids is observed due to the variety of SRO types (see [19, 20] and references cited).

Since SRO is the molecular order formed due to microscopic forces, the correlation length*ξ*SRO is
equal to or exceeds the range of direct molecular interactions. Therefore, to describe heterophase states,
a mesoscopic theory is needed, in which molecular species of size*r* ∼*ξ*SROspeciﬁed by SRO are“ele-
mentary”structural elements rather than molecules. These are not molecular potentials that determine
the equilibrium states and relaxation dynamics of heterophase states but rather the parameters of het-
erophaseﬂuctuation interactions connected with molecular potentials. Evidently, the mesoscopic Hamil-
tonian is more universal but less detailed than the microscopic Hamiltonian speciﬁed by molecular po-
tentials. Parameters of the mesoscopic Hamiltonian can be considered as phenomenological coeﬃcients
with averaged out microscopic details of molecular interaction.

These ideas are in the base of the heterophaseﬂuctuation model (HPFM) [10, 21–27] which is con- sidered in sections 2–6 and in appendixes A and B. It is further used while considering the issues of the thermodynamics of a liquid-glass transition and polymorphous transformations of glass-forming liquids and glasses induced by the SRO reordering and mutual ordering of heterophaseﬂuctuations (section 7 and appendix C). The cooperative relaxation dynamics of a heterophase liquid is considered within the framework of phenomenological model formulated in HPFM [10, 22] (section 8). Conclusive remarks are placed in section 9.

**2. Hetrophase ﬂuctuations and the order parameter**

The heterophaseﬂuctuation is an embryo of a foreign phase in the matrix phase. In many liquids,
even in normal state (above the crystallization temperature,*T**m*), solid-like species are revealed by means
of difractometry. Theﬁrst observation of such heterophaseﬂuctuations (HPF) was made by Stewart and
Morrow [28]. They have discovered sybotactic groups (transient molecular solid-like clusters possessing
speciﬁc short-range order) in simple alcohols above*T**m*.

The HPF are non-perturbativeﬂuctuations in contrast to perturbativeﬂuctuations of physical quan-
tities near their equilibrium values in the homophase state^{3}. Theory of the heterophase states originates
from Frenkel’s paper [29]. Frenkel has coined the term“heterophaseﬂuctuations”and explored the ther-
modynamics of heterophase states ofﬂuid and gas in the vicinity of the phase coexistence curve. Frenkel’s
theory is applicable to all kinds of the coexisting phases (including theﬂuid and solid phases) far below
the critical point. In this case, the amount of substance belonging to HPF is small, and thus Frenkel’s
droplet model, with non-interacting nuclei of a foreign phase, properly describes the heterophase state.

Frenkel’s theory fails in the case of strong HPF, when the fraction of molecules belonging to the

“droplets”is large (for example, when it is near or exceeds the percolation threshold), and thus droplet- droplet interaction cannot be neglected. Besides, this theory was not generalized to include in its consid- eration the states with many SRO-types of the nucleating“droplets”. Both these restrictions of the Frenkel model are obviated in the HPFM.

1Survey article [12] is devoted to the physics of heterogeneous glass-forming liquids.

2Just a few of a huge number of papers devoted to this subject are cited.

3Review [30] is a good introduction to the physics of HPF. The role of non-crystalline solid embryos in vitriﬁcation of organic low-molecular substances (e.g., phenols) was discussed by Ubbelohde in [31].

The HPFM is based on the statistics of the transient solid-like andﬂuid-like mesoscopic species (clus-
ters) which are called*s*-ﬂuctuons and*f*-ﬂuctuons, respectively. By deﬁnition, eachﬂuctuon is speciﬁed
by SRO. The minimal size of aﬂuctuon is equal to the SRO correlation length,*ξ*SRO. An arbitrary number
of types of the*s*-ﬂuctuons,*m*Ê1, can be included into consideration.

To escape needless complications, let us assume that the ﬂuctuons are uniform-sized with size*r*0
and with the number of molecules perﬂuctuon equal to*k*0∼*r*_{0}^{3}. Thus,*ξ*SRO≃*r*0. This simpliﬁcation is
reasonable from the physical point of view because in the both states SRO is formed due to the action of
the same microscopic forces, and the difference of the densities of a liquid and a solid usually amounts
to just a few percent. The solid-like andﬂuid-like fractions consist of*s*- and*f*-ﬂuctuons, respectively.

Let us denote by *N* the total number of molecules of liquid and by*N**f*,*N*1, . . . ,*N**m* the numbers of
molecules belonging to*f*- and*s*-ﬂuctuons,

*N**f* +*N*1+. . .+*N**m*=*N*. (2.1)
The total number ofﬂuctuons is*N*_{ﬂuct}=*N*/*k*0.

The(*m*+1)-component order parameter of the heterophase liquid is determined as follows:

{*c*}=(*c**f*,*c*1, . . . ,*c**m*), *c**i*=*N**i*

*N* Ê0, *i*=*f*, 1, . . . ,*m*, (2.2)

*c**f*+*c*1+. . .+*c**m*≡*c**f* +*c**s*=1. (2.3)

Evidently,*c**i*is the probability of the molecule belonging to*i*-th type ofﬂuctuons.*N**s*=*N*1+*N*2+. . .+*N**m*=
*c**s**N*is the number of molecules of the solid-like fraction. The spatial distribution of theﬂuctuons on scale
*r*≫*r*0can be described by the order parameterﬁelds*c**i*(*x*)with mean values equal to*c**i*.

Let us regard the*k*-th type*s*-ﬂuctuons as statistically insigniﬁcant if*c**k*≪*m*^{−}^{1}. The*f*-ﬂuctuons be-
come statistically insigniﬁcant if*c**f* ≪1. The exclusion of the statistically insigniﬁcant components of
the order parameter from consideration allows one to simplify the equations of HPFM. The statistically
insigniﬁcant entities, when necessary, can be included into consideration as perturbations.

**3. The quasi-equilibrium glass transition and “ideal” glass**

Let us consider more in detail the formulated in Introduction conditions under which the glass tran- sition with equilibrated SRO takes place:

1) The liquid cooling time or the observation time,*τ*obs, should be less than the time of crystallization,

*τ*obs≪*τ*LRO, (3.1)

*τ*LROis the time of long-range ordering.

2) The observation time is much longer than the time of short-range order equilibration,

*τ*obs≫*τ*SRO∼*τ**α*. (3.2)

Reordering of SRO due to localized cooperative rearrangement of the molecular structure is an elemen-
tary*α*-relaxation event. Therefore, it is put*τ*SRO=*τ**α*(*τ**α*is the*α*-relaxation time).

The condition (3.1) limits the value of*τ*obsfrom above. The temperature-time-transformation diagram
can be used to estimate*τ*LROand to outline the area on the(*t*,*T*)-plane in which the condition (3.1) is
satisﬁed.

The condition (3.2) restricts the value of*τ*obsfrom below. It implies that the SRO is equilibrated during
the glass formation. Hence, the order parameter (2.3) is a function of*P*and*T* and depends on time*t*just
because*P*and*T* depend on*t*. When this condition is satisﬁed, the glass transition can be considered as
a sequence of quasi-equilibrium transformations of the SRO.

Due to a dramatic increase of*τ**α*with the temperature decrease near*T*g, the condition (3.2) can be
satisﬁed just above*T*g. Evidently, the condition (3.2) cannot be satisﬁed below the temperature*T*F(*τ*obs)
determined as the root of the equation

*τ**α*(*T*)¯

¯_{T}

F=*τ*obs. (3.3)

This is the temperature of kinetic glass transition because below*T*F(*τ*obs)the SRO can be considered as

“frozen”. Glass transition temperature*T*gdetermined from the viscosity measurements or by means of
calorimetry or dilatometry at the same thermal history is usually equal to*T*F(*τ*obs)with good accuracy,
i.e.,*T*g≃*T*F.

In the limiting case, with*τ*obs→ ∞^{and}*τ*obs≪*τ*LRO, when both conditions (3.1) and (3.2) are satis-
ﬁed, the quasi-equilibrium cooling of a liquid leads to the formation of hypothetical“ideal”glass (with
equilibrated SRO and MRO but without any LRO). Hereinafter, the term“ideal glass”is used in this sense.

It is worth to note that due to the condition (3.1), the residual conﬁgurational entropy of the“ideal”

glass is not equal to zero at*T* →0 because any two parts of such a glass can be considered as non-
correlated and statistically independent if the distance between them exceeds the largest correlation
length which isﬁnite by deﬁnition.

In publications, the issues concerning the physical properties of equilibrium amorphous states below
*T*g are often debated. Between them, the hypothetical vanishing and non-analyticity of the conﬁgura-
tional entropy,*S*conf(*T*), as a function of temperature, at aﬁnite temperature*T*K(the Kauzmann paradox)
[32], and Vogel-Fulcher-Tamman singularity of*τ**α*(*T*)at a temperature*T*VFT[33–35] are under discussion.

In the Adam-Gibbs model [36], the Kauzmann“entropy crisis”is included as an assumption which leads
to the VFT relaxation time singularity at*T*K. Thus, in the Adam-Gibbs model*T*VFT=*T*K. The values of*T*K

and*T*VFTfound from theﬁttings of data on thermodynamics and dynamics of many glass-forming liquids
are close,*T*VFT≈*T*K. Due to the above noted absence of the“entropy crisis”in the“ideal”glass, one can
conclude that*T*Kand*T*VFTshould be considered as free parameters of the widely used phenomenological
model [36]. The issue of proximity of*T*Kand*T*VFTis considered and conﬁrmed within the framework of
HPFM in [37].

**4. Mesoscopic free energy of the heterophase liquid**

The phenomenologic free energy of the heterophase liquid in terms of the introduced order parame-
ter can be presented in the form of polynomial expansion in powers of{*c**i*(*x*)},

*G*(*P*,*T*; {*c*(*x*)})=*G*L(*P*,*T*)+*G*V(*P*,*T*) . (4.1)

In the summand*G*L(*P*,*T*), just local interactions of theﬁelds{*c**i*(*x*)}are included,
*G*L(*P*,*T*)=

Z

*g*L(*x*,*P*,*T*)d^{3}*x*, (4.2)

*g*L(*x*,*P*,*T*) = X

*i*

*c**i*(*x*)*g*_{i}^{0}(*P*,*T*)+*z*
2

X

*i*,*k*

*c**i*(*x*)*c**k*(*x*)*g*^{0}_{ik}(*P*,*T*)

+*T*X

*i*

*c**i*(*x*) ln*c**i*(*x*)+*g*0(*P*,*T*) . (4.3)

*g*_{i}^{0}(*P*,*T*)is independent of the order parameter free energy of*i*-thﬂuctuon;*g*_{ik}^{0} (*P*,*T*)Ê0is theﬂuctuonic
pair interfacial free energy;*z*is theﬂuctuonic coordination number which is taken as independent of the
ﬂuctuon type.

The summand *G*V(*P*,*T*)describes contribution of non-local (volumetric) interaction of*s*-ﬂuctuons,
which is taken in the following form

*G*V(*P*,*T*)= *N*
*k*0

2*π*X

*i*,*j*

Z

Φ(*r*)*w**i j*(*r*)*r*^{2}d*r*, *r*=¯

¯~*x*−~*x*^{′}¯

¯, (4.4)

*w**i j*(*r*)= 〈*c**i*(*x*)*c**j*(*x*^{′})〉 =*V*^{−}^{1}
Z

*c**i*(*x*)*c**j*(*x*^{′})d^{3}*x*, (4.5)
*w**i j*(*r*)is the pair correlation function of*s*-ﬂuctuons,*V* is the volume,Φ(*r*)is the potential of pair inter-
action of the s-ﬂuctuons. This interaction, analogous to the attraction potential of colloid particles in a
solvent, plays a signiﬁcant role in states with diluted solid-like species because it provides aggregation of

the*s*-ﬂuctuons, leading to the Fischer cluster formation. It is taken as Yukawa potential with cutoff range
*R*0which is larger than but comparable with*r*0,

Φ(*r*)= −*ϕ*

*r* exp(−*r*/*R*0). (4.6)

Fluctuonic short-range correlation appears due to both local and volumetric interactions. The Ornstein-
Zernike (OZ) equation [38] can be used to estimate theﬂuctuonic correlation length,*ξ**f l*. It follows from
OZ equation that far from a critical point,*ξ**f l*is comparable with the correlation length of the direct cor-
relation function, which, in turn, is comparable with the range of theﬂuctuonic pair interaction potential.

With*R*0É2*r*0we have that*ξ**f l*≃2*r*0≃2*ξ*SRO. As it is seen, the ordering ofﬂuctuons causes extension of
the molecular pair correlations beyond*r*0and the formation of the of molecular MRO. The liquid region
of size*ξ**f l* with correlatedﬂuctuons is referred to as correlated domain.

The fact that the components of the order parameter*A**i*(*x*)are normalized probabilities, which can-
not exceed 1, validates the presentation of*G*(*P*,*T*)in the form of the polynomial expansion in powers of
{*c**i*(*x*)}.

The connection of the phenomenological free energy (4.1)–(4.6) with the Gibbs free energy can be found using the approach formulated in [39]. It is shown [39] that the free energy presented in terms of the order parameter plays the role of the eﬃcient Hamiltonian in the Gibbs statistics and determines the most probable state of the system. The interplay between the mesoscopic free energy and the Gibbs statistics is considered in appendix A.

**5. The ﬂuctuon-ﬂuctuon interaction and the frustration parameter**

The physical meaning of the pair interaction coeﬃcients of the neighboringﬂuid-like and solid-like ﬂuctuons is clear. It is theﬂuid-solid interfacial free energy taking into account the geometry of the contactingﬂuctuons.

The solid-like fraction can be considered as a mosaic composed of*s*-ﬂuctuons with different SRO. The
interfacial free energy of a pair of*s*-ﬂuctuons depends on their mutual orientations. Evidently, coher-
ent joints of the non-crystalline*s*-ﬂuctuons is hampered at any orientation. The interfacial free energy
increase due to the geometric badness of theﬁt of contacting*s*-ﬂuctuons is*the structural frustration pa-*
*rameter*^{4}. Because of its importance, let us consider theﬂuctuonic frustration parameter more in detail.

A non-crystalline solid-like cluster grows due to the attachment of new molecules. Hence, the former surface molecules become the inner ones and the non-crystalline cluster structure becomes frustrated because not all newly formed coordination polyhedra are exactly similar to the initial polyhedron. A part of them can have the geometry similar to that of the initial coordination polyhedron but slightly de- formed. The occurrence of the coordination polyhedra of completely different geometry is also possible.

Thus, if the initial coordination polyhedron has some symmetry, the newly formed coordination polyhe- dra have a violated or completely changed symmetry. Consequently, the binding energies of the attached molecules appear smaller than that of the inner molecule.

A decrease of the binding energy per molecule is accompanied by an increase of the conﬁgurational entropy due to ambiguities of the geometrical changes of the new coordination polyhedra.

As an example, let us consider the growth of a*z*-vertex coordination polyhedron in the case when
the addition of a new coordination shell leads to the formation of*z*−1new coordination polyhedra with
similar but deformed initial coordination polyhedron while one of them has a different geometry. In this
case, the energy of the inner*z*+1molecules is

*E**z*+1=*ε*0(*z*+1)+*ε*¯def(*z*−1)+*ε*1=*ε*0(*z*+1)+*ε*frust, (5.1)
*ε*0is the mean energy of the initial cluster,*ε*¯defis the mean energy of deformation and*ε*1is the energy
of a molecule with the coordination polyhedron of different geometry. The last two terms in r.h.s. of

4For more information on the structural frustration see e.g., [40] and references cited. The importance of the frustration param- eter at glass transition was considered and discussed qualitatively in [41, 42]. A speciﬁc frustration parameter avoiding the critical point is introduced in the model of frustration-limited domains (FLD) [43, 44].

(5.1) determine the frustration energy,*ε*frust. Because of uncertainty of the last molecule position, the
frustration conﬁgurational entropy due to this uncertainty is as follows:

*s*frust=*s**z*+1=*k*_{B}ln*z*. (5.2)

The frustration free energy is as follows:

*g*frust=*ε*frust−*T s*frust. (5.3)

As it is seen,*ε*frustis∼*z*while*s*frust∼ln*z*. Therefore,*g*frust>0with*z*≫1.

One can conclude that generally the structure of interfacial layer of contactingﬂuctuons is frustrated
and that*g*frust>0.

**6. Equations of the liquid state equilibrium**

Variation of the free energy functional (3.1) at condition (2.3) yields the equations of equilibrium state,
*δ*

*δc**i*(*x*)*G*(*P*,*T*)+*λ* *∂*

*∂c**i*(*x*)
X

*k*

*c**k*(*x*)=0, (6.1)

*λ*is the Lagrange multiplier.

Let us denote by*µ**i*(*P*,*T*)the derivative
*µ**i*(*P*,*T*) = *∂*¡

*g**l*+*g**v*¢

*∂c**i* =*g*_{i}^{0}+X

*k*

*c**k*(*x*)*g**ik*+*T*ln*c**i*(*x*)

+X

*j*

Z

Φ(*x*,*x*^{′})*c**j*(*x*^{′})d^{3}*x*^{′}. (6.2)

Here,*g**ik*=*zg*_{ik}^{0}. Variables(*P*,*T*)are not shown.

As a result, it follows from (6.1) that

*µ**f*(*P*,*T*)=*µ*1(*P*,*T*)=. . .=*µ**m*(*P*,*T*)= −*λ*. ^{(6.3)}
These equations are analogous to the Gibbs equations of the equilibrium of phases.

Equilibrium state is stable if the quadratic form

°

°

°

*δ*^{2}*G*
*δc**i**δc*_{k}

°

°

°is positively deﬁnite.

**7. Solutions of the equations of state**

**7.1. Two-state approximation**

In the physics of glass-forming liquids, different two-state models are in use for a long time [43, 45–57].

HPFM in the two-state approximation provides abbreviated entry of the glass transition.

In fact, in the two-state approximation of the HPFM, the mesoscopic substructure of the solid-like
fraction is neglected and the order parameter in the two-state approximation has just two components,
*c**s*and*c**f*,

*c**s*+*c**f* =1. (7.1)

Applying the spatial averaging, we obtain from (6.2)–(6.3)
(1−2*c**s*) ˜*g**s f*+*T*ln *c**s*

1−*c**s* =*h**s f*. (7.2)

Here,

˜

*g**s f* =*g**s f*−*g**ss*/2;*h**s f* =*g*^{0}_{f} −*g*_{s}^{0}−*g**ss*/2, (7.3)
*g*_{s}^{0}=X

*k*

*c*_{k}^{∗}*g*_{k}^{0}+*T*X

*k*

*c*_{k}^{∗}ln*c*^{∗}_{k}, *g**ss*=X

*g**ik**c*_{i}^{∗}*c*^{∗}_{k}, *c*^{∗}_{i} =*c**i*/*c**s*, (7.4)

*g**ss* is the frustration parameter. It depends on the interaction coeﬃcients of the*s*-ﬂuctuons and proba-
bilities©

*c*_{i}^{∗}ª

. For a while, the volumetric interactions (4.6) are not accounted for.

In the two-state approximation, the coeﬃcient*g**s f* and the frustration parameter*g**ss*are taken as con-
stants. Some remarks concerning the accuracy of two-state approximation of HPFM appear in section 9.

Equation (7.2) is isomorphic to the equation of state of the Ising model with an externalﬁeld*h**s f*. The
solution of equation (7.2) at*c**s*≪1is as follows:

*c**s*(*T*)=exp©£

∆*s**f s*

¡*T*_{e}^{0}¢ ¡
*T*_{e}^{0}−*T*¢

−*g**s f*

¤/*T*_{e}^{0}ª

. (7.5)

Here,

∆*s**f*,*s*(*T*)= −

*∂*³
*g*^{0}_{s}−*g*^{0}_{f}´

*∂T* =*s**f*(*T*)−*s**s*(*T*) (7.6)

is the difference of entropies of the*f*- and*s*-ﬂuctuon.*T*_{e}^{0}is the solution of the equation
*g*^{0}_{f}¡

*P*,*T*_{e}^{0}¢

=*g*_{s}^{0}¡
*P*,*T*_{e}^{0}¢

. (7.7)

At*c**f* =1−*c**s*≪1

*c**f*(*T*)=exp©£

∆*s**f*,*s*

¡*T*_{e}^{1}¢ ¡
*T*_{e}^{1}−*T*¢

−*g**s f*

¤/*T*_{e}^{1}ª

, (7.8)

where*T*_{e}^{1}is the solution of the equation
*g*^{0}_{f} ¡

*P*,*T*_{e}^{1}¢

=*g*_{s}^{0}¡
*P*,*T*_{e}^{1}¢

+*g**ss*. (7.9)

The physical meaning of the characteristic temperatures*T*_{e}^{0},*T*_{e}^{1}is explained below.

The temperature*T**e*, at which the“externalﬁeld”*h**s f* is equal to zero, is the coexistence temperature
of two heterophase liquid states determined by equation

*g*^{0}_{f} (*P*,*T**e*)=*g*_{s}^{0}(*P*,*T**e*)+*g**ss*/2. (7.10)
At*T*=*T**e*, we have*c**s*(*T**e*)=*c**f*(*T**e*)=1/2. In the vicinity of*T**e*,

*c**s*≈1

2+ *h**s f*(*T*)
2(2*T**e*−*g*˜*s f*)

"

1−

2*T h*^{2}_{s f}(*T*)
3(2*T**e*−*g*˜*s f*)^{3}

#

=1

2+∆*s**s*,*f*(*T**e*) (*T*−*T**e*)
2(2*T**e*−*g*˜*s f*) +*O*¡

(*T*−*T**e*)^{3}¢

. (7.11)

As it follows from (7.7), (7.9) and (7.10),

*T*_{e}^{0}≈*T**e*+*g**ss*/2∆*s**f*,*s*, *T*_{e}^{1}≈*T**e*−*g**ss*/2∆*s**f*,*s*. (7.12)
The solution (7.11) is stable at*g*˜*s f*(*P*,*T**e*)<2*T**e*. If*g*˜*s f*(*P*,*T**e*)>2*T**e*, it is unstable and at*T*=*T**e*, (*P*)theﬁrst
order phase transition takes place.

0.0 0.2 0.4 0.6 0.8 1.0

0.6 0.8 1.0 1.2

Temperature, T/T 0

e

c s

T 0

e T

e

a

T 1

e

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0.0

0.5 1.0

b

T 0

1

c s

Temperature, T/T 0

e T

e T

1

e

**Figure 1.**(Color online) The solid-like fraction of of liquid,*c**s*, vs*T*/*T*_{e}^{0}at (a)*g*˜_{s f}(*P*,*T**e*)>2*T**e*and (b)

˜

*g*_{s f}(*P*,*T*_{e})<2*T*_{e}.

**Figure 2.**(Color online) Schematic representation of the hetrophase liquid states: (a) rare*s*-ﬂuctuons
inﬂuid described by equation (7.5); (b) *f*-ﬂuctuons in glass [equation (7.8)]; (c) heterophase state with
comparable fractions of the*s*- and*f*-ﬂuctuons [equation (7.11)].

Graphic representation of solutions of equation (7.2) is shown inﬁgure 1. The stable and unstable solutions are depicted by solid and dashed lines, respectively.

If*g*˜*s f* =2*T**e*, i.e.

*g**ss*=2*g**s f*−4*T**e*, (7.13)

then, the 2nd order phase transition takes place at*T*=*T**e*(*P*). In accordance with (7.5) and (7.8), above*T*_{e}^{0}
and below*T*_{e}^{1}, the HPF are weak but within the temperature range£

*T*_{e}^{1},*T*_{e}^{0}¤

, where in compliance with
(7.11)*c**s*and*c**f* are comparable quantities, they are strong.

It worth to note that the solutions (7.5) and (7.8) reproduce the results of Frenkel’s model in the vicin-
ity of the phase coexistence temperatures (here,*T*_{e}^{0}and*T*_{e}^{1}, respectively). Therefore,*T*_{e}^{0}can be consid-
ered as the coexistence temperature of theﬂuid and heterophase liquid phases while*T*_{e}^{1}is the phase
coexistence temperature of the“ideal”glass (as it is determined above) and the heterophase liquid. Thus,
*T*_{e}^{1}is the ideal glass transition temperature. The real glass transition temperature,*T*g, which depends on
*τ*obs(see section 3), is above*T*_{e}^{1}due to dramatic retarding of the structure relaxation with temperature
decrease. For this reason, the real glass transition temperature range,£

*T*g,*T*_{e}^{0}¤

, is narrower than£
*T*_{e}^{1},*T*_{e}^{0}¤

.
The structure of the heterophase states in the vicinity of the characteristic temperatures*T*_{e}^{1},*T**e* and*T*_{e}^{0}
is schematically presented inﬁgure 2. Inﬁgure 3, the mesoscopic structure of the solid-like fraction with
several types of*s*-ﬂuctuons is shown schematically. Let us remind that the solutions of equations (7.5),
(7.8), (7.11) are obtained under the assumption that the fractions of*s*-ﬂuctuons{*c**i*} ,*i*=1, . . . ,*m*are nearly
constant or they are changing continuously and smoothly. This assumption fails if a phase transformation
with stepwise changes of the fractions{*c**i*}within the solid-like fraction takes place. In the next section,
the impact of such a phase transformation within the solid-like fraction on the features of theﬂuid-solid

**Figure 3.**(Color online) The same as inﬁgure 2 (c) but the mesoscopic structure of the solid-like fraction
containing several types of the*s*-ﬂuctuons is shown.

phase transformation is considered.

**7.2. Phase transition in the solid-like fraction**

Evidently, a phase transition in the solid-like fraction causes a non-analytic behaviour of the solutions
of equation (7.2). This type of the liquid-liquid transition appears due to multiplicity and interaction of
the*s*-ﬂuctuons which leads to the mutual ordering and phase separations within the solid-like fraction.

As a minimal model, let us consider the heterophase liquid with two types of *s*-ﬂuctuons. Hence,
*m*=2. Thus, in (6.3)*i*,*j*=1, 2. The equation of state (6.3) for the solid-like fraction is as follows:

¡1−2*c*_{1}^{∗}¢

*c**s**g*˜12+*T*ln *c*_{1}^{∗}

1−*c*_{1}^{∗}=*h*12, *c*^{∗}_{i} =*c**i*/*c**s*, *c*_{1}^{∗}+*c*_{2}^{∗}=1 ,

*g*˜12=*g*12−¡

*g*12+*g*22¢

/2, *h*12=*g*_{2}^{0}−*g*_{1}^{0}+*c**s*¡

*g*22−*g*11¢

/2. (7.14)

It is seen that this equation is isomorphic to equation (7.2) but the“externalﬁeld”*h*12and the pair
interaction coeﬃcient*c**s**g*˜12 depend on*c**s*. Therefore, associated solutions of equations (7.2) and (7.14)
should be considered together. The search for a general solution of these nonlinear equations at an arbi-
trary set of coeﬃcients is a cumbersome and hardly attractive task because the values of the coeﬃcients
for substances are initially unknown. Nevertheless, we can look for some“typical”solutions at a reason-
able speciﬁcation of the coeﬃcients.

As a useful example, let us consider solutions of equation (7.14) in the vicinity of the coexistence
curve,*h*12(*P*,*T*)=0, assuming that *c**s*¡

*g*22−*g*11¢

is a negligible quantity. In this case, the coexistence
temperature,*T*12, is determined by equation

*g*_{1}^{0}(*P*,*T*12)=*g*_{2}^{0}(*P*,*T*12) . (7.15)
It is assumed that*T*12is above the coexistence temperature*T**e*. A phase transformation of the solid-
like fraction and induced liquid-liquid phase transition at*T*12<*T**e*is considered in [27].

In the vicinity of*T*12

*h*12=*g*_{1}^{0}(*P*,*T*)−*g*_{2}^{0}(*P*,*T*)≈(*s*1−*s*2) (*T*−*T*12)≡∆*s*12(*T*−*T*12) , (7.16)
*s*1and*s*2is the entropy of*s*-ﬂuctuon of type 1 and 2, respectively.

If *c**s*(*T*12) ˜*g*12>2*T*12, then there are two stable solutions of equation (7.14) and, as a consequence,
two stable solutions of equations (7.2), (7.3). Consideration of these solutions is contained in appendix B.

Their graphic representation is shown inﬁgure 4. The jump of the parameter*c**s*at*T*=*T*12is [see (B.7) in
appendix B] as follows:

∆*c**s*(*T*12)≈(*s*1−*s*2)exp¡

−*g**s f**β*¢

. (7.17)

0 a

T c,2

C

*1

Temperature, T/T 0

1 1 1

T 12

T c,1

1.0 1.2

0.0 0.1 0.2 0.3 0.4 0.5

b

Solid-likefraction,cs

Temperature, T/T c T

12

**Figure 4.**(Color online) First order phase transition within the solid-like fraction (a) induces the liquid-
liquidﬁrst order phase transition (b).

a

C

* 1

Temperature T/T 1,0

e 1

1 T

12

0.0 0.1 0.2 0.3 0.4 0.5

b

Solid-likefraction,cs

Temperature, T/T c T

12 1

**Figure 5.**Continuous but stepwise evolution of*c*_{1}^{∗}(*T*)shown in a) induces the appearance of the inﬂection
point of*c**s*(*T*)at*T*=*T*_{12}, b).

If *c**s*(*T*12) ˜*g*12<2*T*12, the fractions*c*_{1}^{∗},*c*^{∗}_{2} and parameter*c**s* change continuously at*T* =*T*12but the
inﬂection points of the functions*c*_{1}^{∗}(*T*)and *c**s*(*T*)appear at*T* =*T*12 (ﬁgure 5). Hence, the expression
(7.17) estimates the bench height of the parameter*c**s*.

The heat of the phase transitions is determined by equation (B.8) in appendix B,

∆*H*12≃*c**s*(*T*12) (*s*1−*s*2)*k*_{0}^{−}^{1}£

*H**f s*+*T*12

¤, (7.18)

*H**f s* is the heat of theﬂuid-solid phase transition. It is taken into account here that the heat of 1↔2
solid-solid phase transition is equal to(*s*1−*s*2)*k*_{0}^{−}^{1}*T*12.

It is worth to note that at aﬁxed value of*c**s*, equation (7.14) is isomorphic to Ising model with non-
zero externalﬁeld and with exchange integral*c**s**g*˜12which can be positive or negative. The externalﬁeld
controls the ratio*c*^{∗}_{1}/*c*^{∗}_{2}while the sign of the exchange integral determines the type of mutual ordering of
the*s*-ﬂuctuons. With*c**s**g*˜12>0(ferromagnetic interaction),*s*-ﬂuctuons of different types tend to separate.

At*c**s**g*˜12<0, the“antiferromagnetic”order with alternating*s*-ﬂuctuons of different types is preferable.

In both cases, theﬂuctuonic SRO generates molecular medium-range order with the correlation length
*ξ**f l*∼2*r*0[in compliance with the general conclusion made in section 4 after equation (4.6)].

**7.3. The Fischer cluster**

Along with the above considered types of MRO appearing due to localﬂuctuonic interaction, there
is a different type of theﬂuctuonic order with comparatively large (as it was observed, up to∼102nm)
correlation length,*ξ*FC≫*ξ**f l*. It appears due to the aggregation of the*s*-ﬂuctuons under the effect of the
volumetric gravitation potential (4.6). The equilibrated aggregation of*s*-ﬂuctuons possesses the fractal
structure with fractal dimension, correlation length, equilibration time and relaxation dynamics depend-
ing on the liquid features and temperature. This remarkable phenomenon, which was discovered and
investigated in detail by Fischer et al. [1–10, 23–27], is known as the Fischer cluster. The Fischer cluster
was visualized by observing a speckle pattern in ortho-terphenil [6]. The speckle patternﬂuctuates and
rearranges very slowly, with characteristic time∼1min at*T* =293K, while the*α*-relaxation time at
this temperature is*τ**α*=40ns. With temperature increase, the speckle size and contrast decreases and
at*T* >340K no speckle is seen. Schematically, the heterophase liquid structure with and without the
Fischer cluster is shown inﬁgure 6. It is worth to note that the Fischer cluster formation in heterophase
liquid is not an exclusion but the rule if the Fischer cluster equilibration time,*τ*FC, is shorter than the
observation time, i.e., if*τ**α*≪*τ*FC≪*τ*obs≪*τ*LRO. The heterogeneous structure and slow structural re-
laxation are observed not only in many Van der Waals molecular liquids but also in some metallic melts
above*T**m*[58–60].

The Fischer cluster was originally identiﬁed using the results of the small-angle X-ray scattering on the densityﬂuctuations. The conventional large-scale densityﬂuctuations in a homophase liquid are

**Figure 6.**(Color online) Schematic fragments of the heterophase liquid with (a) and without (b) Fischer’s
cluster. Just two types of statistically signiﬁcant*s*-ﬂuctuons are shown. The circle shows the size of the
correlated domain.

proportional to the isothermal compressibility*κ**T* and are independent of the wave vector*q*,

¯

¯*ρ*(*q*)¯

¯

2|*q*→0∼*κ**T**T*. (7.19)

Here,*q*is the wave vector,*ρ*(*q*)is the amplitude of densityﬂuctuations. The intensity of X-ray scattering
on the densityﬂuctuations,*I*¡

*q*¢

, is proportional to¯

¯*ρ*(*q*)¯

¯

2.
It appears that a*q*-dependent excess scattering intensity,*I*exc

¡*q*¢

∼*q*^{−}^{D} (*D*is the fractal dimension)
occurs at*T*<*T**A*≈*T*_{e}^{0}. The*I*exc

¡*q*¢

is much larger than the scattering intensity on the thermalﬂuctuations
(7.19). The results of the wide-angle X-ray scattering show that SRO of the liquid contains both theﬂuid-
like and solid-like components at*T* <*T**A*[9, 61]. It turns out that the thermodynamics and*α*-relaxation
dynamics are quite the same in the liquid states with and without the Fischer cluster. It means that the
changes of the thermodynamic properties due to the Fischer cluster formation are too small to be reliably
detected and that theﬂuctuonic SRO does not undergo noticeable changes. Therefore, the Fischer cluster
formation can be considered as the process of self-organization of the correlated domains (CDs), i.e.,
entities possessing theﬂuctuonic SRO with the correlation length*ξ**f l*≈2*r*0, (section 4).

Theory of the Fischer cluster is developed in [10, 23–26]. The Fischer cluster is considered as a fractal
aggregation of*s*-ﬂuctuons with the fractal dimension*D**f* and correlation length*ξ*FC. Minimization of the
free energy as function of*c*¯*s*,*D**f* and*ξ*FCallows one to determine the equilibrium values of*D**f* and*ξ*FC.
It is found (see appendix C) that

*ξ*FC

¡*c**s*,*D**f*

¢≈¡
*c**s*

¢

1

*D f*−3*r*0, *D**f* =3−

µ
ln *r*0

*ξ**f l*

¶_{−}1

ln*c**s*,CD, (7.20)

*c**s*,CDis the concentration of*s*-ﬂuctuons within CD ([24–26], appendix C)

*c**s*,CD=

"

1+ *I*(*D**f*,*ξ*FC)

*∂*^{2}*g*^{0}¡
*c**s*

¢/*∂c*^{2}_{s}−*I*(*D**f*,*ξ*FC)

#

*c**s*Ê*c**s*. (7.21)

The quantity*I*(*D**f*,*ξ*), (C.21), is proportional to*φ*0. Equation (7.21) shows that within the CD, concentra-
tion of*s*-ﬂuctuons is larger than its mean value*c**s*:*c**s*,CD≃*c**s*+^{const}*φ*0.

The liquid state with the Fischer cluster is stable (while the state without the Fischer cluster is metastable or unstable) at

*c**s*>*c**s*,0<

µ*r*0

*ξ**f l*

¶2

≈0.16, 1<*D**f* É3. (7.22)

Transformation of the state without Fischer’s cluster into the state with Fischer’s cluster is a weak
ﬁrst order phase transition with the transformation heat∝*ϕ*0.

The upper bound of the*c**s*-range in which the Fischer cluster exists,*c**s*,1=(*ξ**f l*/*r*0)^{2}*c**s*,0 [see (C.26)],
decreases,∼*φ*0, with an increase of the strength of the*s*-ﬂuctuons gravitation potential*φ*0. When *c**s*

**Figure 7.**(Color online) Parametric phase diagram on the plane¡
*T*^{∗},*g*^{∗}_{ss}¢

,*T*^{∗}=*T*/*T*_{e}^{0},*g*_{ss}^{∗}=*g**ss*/∆*s*_{s,f}*T*_{e}^{0}.
The phase coexistence lines*T*=*T*_{e}^{0};*T*=*T**e*;*T*=*T*_{e}^{1}and the threshold of the Fischer cluster formation
temperature,*T*_{FC}^{∗}, approximately determined using equations (7.5), (7.22) are shown. The critical end
point at*g*_{ss,}^{∗}_{c}exists on the line*T*=*T**e*. Bold line 1 schematically presents an evolution phase curve of the
equilibrium system.

approaches*c**s*,1<1from below, the fractal dimension*D**f* approaches 3 and*ξ*FC→ ∞. It means that at
*c**s*Ê*c**s*,1the solid-like fraction consists of the connected 3-dimensional solid-like clusters of size*r*>*ξ**f l*.
Thus, at*c**s*Ê*c**s*,1, the topology of the heterophase liquid equilibrated on scale*r*Ê*ξ**f l*changes.

**7.4. Parametric phase diagram**

In the HPFM, the structure and phase states of the heterophase liquid are described in terms of*T*
and coeﬃcients*g**s f*,*g**ss*,*h**s f*. It is useful to construct a phase diagram (*the parametric phase diagram*) of
the glass-forming liquid in terms of these parameters^{5}. The parametric phase diagram of the two-state
approximation is determined by equations (7.7), (7.9), (7.10) and the equation (7.22) in combination with
(7.5). Namely, they determine the coexistence temperatures of different states in terms of the coeﬃcients
*g**s f*,*g**ss*,*h**s f*. The quantities*T*_{e}^{0},*T*_{e}^{1},*T**e* and*T*FCare coexistence temperatures of

1) theﬂuid and heterophase liquid (*T*_{e}^{0});

2) the heterophase liquid and“ideal”glass(*T*_{e}^{1});

3) theﬂuid-like and solid-like states (*T**e*);

4) the heterophase liquid with and without the Fischer cluster (*T*FC).

Introducing the scaled temperature,*T*^{∗}=*T*/*T*_{e}^{0}, and the frustration parameter*g*_{ss}^{∗}=*g**ss*/∆*s**f*,*s**T*_{e}^{0}, we can
present the relations (7.8) in a dimensionless form,

*T*_{e}^{0}^{∗}=1, *T*_{e}^{∗}=1−*g*_{ss}^{∗}/2, *T*_{e}^{1}^{∗}≈1−*g*_{ss}^{∗}. (7.23)
The end critical point location on theﬂuid-solid phase coexistence curve,*T*^{∗}=*T*_{e}^{∗}(*P*)is located at

*g*^{∗}_{ss,}_{c}=2*g**s f*

± ¡∆*s**s*,*f**T*_{e}^{0}¢

−4*T**e*^{∗}(*P*)>0. (7.24)

Theﬁrst orderﬂuid-solid phase transition on the phase coexistence curve takes place at*g*_{ss}^{∗}<*g*_{ss,}^{∗}_{c}.
The parametric phase diagram depicted on the plane¡

*T*^{∗},*g*_{ss}^{∗}¢

using relations (7.23), (7.24) and (7.22),
(7.5) is shown inﬁgure 7. As an example, here is also shown one phase trajectory which becomes non-
physical below the glass transition temperature*T*_{g}^{∗}. Within the range0<*g*_{ss}^{∗}<*g*^{∗}_{ss,}_{c}theﬁrst order phase
transition takes place on the phase coexistence line*T*^{∗}=*T*_{e}^{∗}. A weakﬁrst order phase transition takes
place on the line*T*^{∗}=*T*_{FC}^{∗}.

5Tentative phase diagrams of glass-forming liquid in terms of the model coeﬃcients are introduced in [41] and then in [62].

**7.5. Static structure factor and the order parameter restoration**

Pair correlation function of the densityﬂuctuations ,^{⌢}*̟*(*q*,*T*), of the heterophase liquid with the Fis-
cher cluster is∼*q*^{−}^{D}at*r*0≪*q*^{−}^{1}≪*ξ*^{−}^{1}. At*qr*0∼1, it is a superposition of the pair correlation functions
ofﬂuctuons,

⌢*̟*(*q*,*T*)=*c**f*(*T*)*̟**f*(*q*)+

*m*

X

*i*=1

*c**i*(*T*)*̟**i*(*q*)=[1−*c**s*(*T*)]*̟**f*(*q*)+*c**s*(*T*)*̟**s*(*q*), (7.25)

*̟**s*(*q*)= 1
*c**s*

*m*

X

*i*=1

*c**i**̟**i*(*q*), (7.26)

*̟**f*(*q*),*̟**i*(*q*)are Fourier transforms of the pair correlation functions of the *f*- and*s*-ﬂuctuons, respec-
tively. The cross-correlation terms∼*c**f**c**s* are omitted in (7.25). The quantities*̟**f*(*q*),*̟**i*(*q*)weakly de-
pend on the temperature. For this reason, the equation (7.25) can be applied to restore the order param-
eter*c**s*(*T*)using the structure factors*̟**f*(*q*),*̟**s*(*q*)measured in the liquid,ﬂuid and glassy states [61]. On
the other hand, as it is shown in [22],*c**s*(*T*)can be restored from calorimetric data using the relation

*c**s*≃£

*H**f*(*T*)−*H*exp(*T*)¤
/£

*H**f*(*T*)−*H**s*(*T*)¤

(7.27)
which follows from equation (4.3). Here,*H**f*(*T*),*H**s*(*T*)are enthalpies of theﬂuid and glass extrapolated
in the temperature range£

*T*_{e}^{1},*T*_{e}^{0}¤

, and*H*exp(*T*)is the experimentally measured enthalpy of the glass-
forming liquid. Comparison of the results of the order parameter restoration from the structural data,
using equation (7.25), and from the calorimetric data, using relation (7.27), gives a good chance to check
the reliability of the HPFM. This procedure was performed using structural and calorimetric data of salol
[9, 61]. Results are presented inﬁgure 8 by scattered symbols. Solution of the equation of state in the two-
state approximation (subsection 7.1), in which the experimentally measured thermodynamic parameters
and free parameter*g*˜*s f* are used, is presented there by a solid line.

Let us remind that the analytic solution describes the order parameter*c**s*(*T*)of the equilibrated sys-
tem. Therefore, it noticeably deviates from the experimentally determined values*c**s*(*T*)near the glass
transition temperature, where the liquid becomes non-equilibrium. Relations (7.25) and (7.27), obtained
without the assumption that the system is equilibrated, allow us to recover the thermal history of“true”

(in the phenomenological sense) value of*c**s*.

**Figure 8.**(Colo online) The solid-like fraction of salol vs*T* as it is found from the analysis of the calori-
metric data (triangles), and from the temperature dependence of the structure factor (circles) [22, 61].

Arrows indicate the temperatures*T*_{e}^{0},*T**e*,*T*g,*T*_{e}^{1}. Line presents an analytic solution of the equation of
state in the two-state approximation withﬁtting parameter*g*˜_{s f}.

**8. Dynamics**

**8.1.** **α** **-relaxation**

**α**

Thermally activated cooperative structural rearrangements which can involve up to∼10^{2}molecules
[19, 20, 63–67] are called*α*-relaxation. A large amount of the molecules are involved in the rearrange-
ment due to correlations. Structural rearrangement of aﬂuctuon also involves rearrangements of the
neighboringﬂuctuons within CD of size*ξ**f l*. Therefore, the size of cooperatively rearranging domain is
nearly equal to*ξ**f l*.

The activation energy of*α*-relaxation,

*E*ac=*d*ln*τ**α*

*dβ* , *β*=1

*T*, (8.1)

depends on the order parameter. It can be presented as an expansion in powers of the order parameter
[10, 22]^{6},

*E*ac=*E*^{0}_{ac}+*E*_{ac}^{1}*c**s*+*E*_{ac}^{2}*c*^{2}_{s}+. . . . (8.2)
Above*T**A*, the activation energy is nearly equal to*E*^{0}_{ac}. Cooperativity of the liquid dynamics is induced by
the*s*-ﬂuctuons interaction which becomes considerable below*T**A*.

Fischer and Bakai [22] have suggested that CD can be rearranged when all the molecules therein are
inﬂuid-like state with correlations destroyed on the scale*ξ**f l*. This assumption leads to the following
expression [22]

*E*ac= *A*

(1−*T*K/*T*)^{2}+*z*CD*c**s*

¡*H**f* −*H**s*

¢+*O*¡
*c*^{2}_{s}¢

. (8.3)

*z*CD∼¡
*ξ**f l*/*a*¢3

is the cooperativity parameter, which is the mean n umber of molecules within the CD;*H**f*,
*H**s*is the enthalpy of liquid-like and solid-like fraction per molecule. Theﬁrst term is taken in the form
proposed for random packings of spheres in [68, 69]. Its denominator takes into account the decrease of
the free volume of theﬂuid and the numerator is equal to the activation energy above*T**A*. The Kauzmann
temperature,*T*K, is aﬁtting parameter (see comments concerning*T*Kin section 3)

**Figure 9.**The activation energy of salol vs the reciprocal temperature [2].

Enthalpies*H**f*(*T*)and*H**s*(*T*)within the temperature range£
*T*g,*T**A*¤

are understood as extrapolations
of these functions measured at*T* >*T**A*and*T*<*T*g, respectively.

As an example of using the equation (8.3) [22], the activation energy of salol was analyzed in [22]. The
activation energy of salol vs the reciprocal temperature is shown inﬁgure 9. The experimental data are
shown by circles. The curve is a result ofﬁtting the formula (8.3) using parameters*A*=967K;*T*K=153K,

6There is no reason to believe that*E*ac(*c**s*)is a singular function at*c**s*∈[0,1].