2. Hetrophase ﬂuctuations and the order parameter

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http://www.icmp.lviv.ua/journal

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 approach. The Fischer cluster phenomenon, glass transition, liquid-liquid transformations, parametric phase diagram, 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,Tm, 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 amorphous (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 cannot 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 scaleξ[let us denote this time byτ(ξ)] and no signiﬁcant structural correlation occurs on scalesr>ξ, one can consider the glass transition as a sequence of transformations of the structure states which are equilibrated just on the scalesr<ξ. Statistical description of such a liquid can be developed if we exclude from the statistics the states with the correlation scaler>ξand, on the other hand, ensure that the observation time,τobs, is much longer thanτ(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 lengthsr>ξleads to an increase of the free energy of the equilibrium state. The standard Gibbs statistics restores withξ→ ∞. The observation time limits from above the scale of the relaxation timeτ(ξ)and, consequently, the scaleξ, becauseτ(ξ) increases∼ξ^θ(the exponentθdepends on the features of the relaxation kinetics).

The spatial scale of the SRO,ξSRO, is minimal among the possible correlation lengths in the liquid. Ac- cordingly,τ(ξSRO)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≫ξSRO, which involves a large number of molecules in rearrangement and is driven by relatively weak multi-molecular forces, takes much longer thanτ(ξSRO)time. The liquid or glass, which is equilibrated on the scaler∼ξSROwithout 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^thbirthday.

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It is experimentally established that the glass-forming liquids are heterophase (their structure consists of the mutually transforming ﬂuid-like and solid-like substructures). Observations of the heterophase 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 others¹.

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 lengthξSRO 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 sizer ∼ξSROspecified 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 heterophasefluctuation interactions connected with molecular potentials. Evidently, the mesoscopic Hamil- tonian is more universal but less detailed than the microscopic Hamiltonian specified by molecular potentials. 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 considered 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,Tm), 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 aboveTm.

The HPF are non-perturbativefluctuations in contrast to perturbativefluctuations of physical quantities near their equilibrium values in the homophase state³. Theory of the heterophase states originates from Frenkel’s paper [29]. Frenkel has coined the term“heterophasefluctuations”and explored the thermodynamics 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 consideration 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].

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The HPFM is based on the statistics of the transient solid-like andfluid-like mesoscopic species (clusters) which are calleds-fluctuons andf-fluctuons, respectively. By definition, eachfluctuon is specified by SRO. The minimal size of afluctuon is equal to the SRO correlation length,ξSRO. An arbitrary number of types of thes-fluctuons,mÊ1, can be included into consideration.

To escape needless complications, let us assume that the fluctuons are uniform-sized with sizer0 and with the number of molecules perfluctuon equal tok0∼r₀³. Thus,ξSRO≃r0. 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 ofs- andf-fluctuons, respectively.

Let us denote by N the total number of molecules of liquid and byNf,N1, . . . ,Nm the numbers of molecules belonging tof- ands-ﬂuctuons,

Nf +N1+. . .+Nm=N. (2.1) The total number ofﬂuctuons isN_ﬂuct=N/k0.

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

{c}=(cf,c1, . . . ,cm), ci=Ni

N Ê0, i=f, 1, . . . ,m, (2.2)

cf+c1+. . .+cm≡cf +cs=1. (2.3)

Evidently,ciis the probability of the molecule belonging toi-th type offluctuons.Ns=N1+N2+. . .+Nm= csNis the number of molecules of the solid-like fraction. The spatial distribution of thefluctuons on scale r≫r0can be described by the order parameterfieldsci(x)with mean values equal toci.

Let us regard thek-th types-fluctuons as statistically insignificant ifck≪m⁻¹. Thef-fluctuons become statistically insignificant ifcf ≪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 transition with equilibrated SRO takes place:

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

τobs≪τLRO, (3.1)

τLROis the time of long-range ordering.

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

τobs≫τSRO∼τα. (3.2)

Reordering of SRO due to localized cooperative rearrangement of the molecular structure is an elemen- taryα-relaxation event. Therefore, it is putτSRO=τα(ταis theα-relaxation time).

The condition (3.1) limits the value ofτobsfrom above. The temperature-time-transformation diagram can be used to estimateτLROand to outline the area on the(t,T)-plane in which the condition (3.1) is satisﬁed.

The condition (3.2) restricts the value ofτobsfrom below. It implies that the SRO is equilibrated during the glass formation. Hence, the order parameter (2.3) is a function ofPandT and depends on timetjust becausePandT depend ont. 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 ofταwith the temperature decrease nearTg, the condition (3.2) can be satisﬁed just aboveTg. Evidently, the condition (3.2) cannot be satisﬁed below the temperatureTF(τobs) determined as the root of the equation

τα(T)¯

¯_T

F=τobs. (3.3)

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This is the temperature of kinetic glass transition because belowTF(τobs)the SRO can be considered as

“frozen”. Glass transition temperatureTgdetermined from the viscosity measurements or by means of calorimetry or dilatometry at the same thermal history is usually equal toTF(τobs)with good accuracy, i.e.,Tg≃TF.

In the limiting case, withτobs→ ∞^andτobs≪τLRO, 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 atT →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 Tg are often debated. Between them, the hypothetical vanishing and non-analyticity of the conﬁgura- tional entropy,Sconf(T), as a function of temperature, at aﬁnite temperatureTK(the Kauzmann paradox) [32], and Vogel-Fulcher-Tamman singularity ofτα(T)at a temperatureTVFT[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 atTK. Thus, in the Adam-Gibbs modelTVFT=TK. The values ofTK

andTVFTfound from theﬁttings of data on thermodynamics and dynamics of many glass-forming liquids are close,TVFT≈TK. Due to the above noted absence of the“entropy crisis”in the“ideal”glass, one can conclude thatTKandTVFTshould be considered as free parameters of the widely used phenomenological model [36]. The issue of proximity ofTKandTVFTis 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 parameter can be presented in the form of polynomial expansion in powers of{ci(x)},

G(P,T; {c(x)})=GL(P,T)+GV(P,T) . (4.1)

In the summandGL(P,T), just local interactions of theﬁelds{ci(x)}are included, GL(P,T)=

Z

gL(x,P,T)d³x, (4.2)

gL(x,P,T) = X

i

ci(x)g_i⁰(P,T)+z 2

X

i,k

ci(x)ck(x)g⁰_ik(P,T)

+TX

i

ci(x) lnci(x)+g0(P,T) . (4.3)

g_i⁰(P,T)is independent of the order parameter free energy ofi-thfluctuon;g_ik⁰ (P,T)Ê0is thefluctuonic pair interfacial free energy;zis thefluctuonic coordination number which is taken as independent of the fluctuon type.

The summand GV(P,T)describes contribution of non-local (volumetric) interaction ofs-ﬂuctuons, which is taken in the following form

GV(P,T)= N k0

2πX

i,j

Z

Φ(r)wi j(r)r²dr, r=¯

¯~x−~x^′¯

¯, (4.4)

wi j(r)= 〈ci(x)cj(x^′)〉 =V⁻¹ Z

ci(x)cj(x^′)d³x, (4.5) wi j(r)is the pair correlation function ofs-fluctuons,V is the volume,Φ(r)is the potential of pair interaction 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

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thes-ﬂuctuons, leading to the Fischer cluster formation. It is taken as Yukawa potential with cutoff range R0which is larger than but comparable withr0,

Φ(r)= −ϕ

r exp(−r/R0). (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,ξf l. It follows from OZ equation that far from a critical point,ξf lis comparable with the correlation length of the direct correlation function, which, in turn, is comparable with the range of theﬂuctuonic pair interaction potential.

WithR0É2r0we have thatξf l≃2r0≃2ξSRO. As it is seen, the ordering ofﬂuctuons causes extension of the molecular pair correlations beyondr0and the formation of the of molecular MRO. The liquid region of sizeξf l with correlatedﬂuctuons is referred to as correlated domain.

The fact that the components of the order parameterAi(x)are normalized probabilities, which cannot exceed 1, validates the presentation ofG(P,T)in the form of the polynomial expansion in powers of {ci(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 ofs-fluctuons with different SRO. The interfacial free energy of a pair ofs-fluctuons depends on their mutual orientations. Evidently, coher- ent joints of the non-crystallines-fluctuons is hampered at any orientation. The interfacial free energy increase due to the geometric badness of thefit of contactings-fluctuons isthe structural frustration pa- rameter⁴. 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 deformed. 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 polyhedra 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 az-vertex coordination polyhedron in the case when the addition of a new coordination shell leads to the formation ofz−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 innerz+1molecules is

Ez+1=ε0(z+1)+ε¯def(z−1)+ε1=ε0(z+1)+εfrust, (5.1) ε0is the mean energy of the initial cluster,ε¯defis the mean energy of deformation andε1is 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 parameter 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].

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

sfrust=sz+1=k_Blnz. (5.2)

The frustration free energy is as follows:

gfrust=εfrust−T sfrust. (5.3)

As it is seen,εfrustis∼zwhilesfrust∼lnz. Therefore,gfrust>0withz≫1.

One can conclude that generally the structure of interfacial layer of contactingﬂuctuons is frustrated and thatgfrust>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

δci(x)G(P,T)+λ ∂

∂ci(x) X

k

ck(x)=0, (6.1)

λis the Lagrange multiplier.

Let us denote byµi(P,T)the derivative µi(P,T) = ∂¡

gl+gv¢

∂ci =g_i⁰+X

k

ck(x)gik+Tlnci(x)

+X

j

Z

Φ(x,x^′)cj(x^′)d³x^′. (6.2)

Here,gik=zg_ik⁰. Variables(P,T)are not shown.

As a result, it follows from (6.1) that

µf(P,T)=µ1(P,T)=. . .=µm(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

°

δ²G δciδc_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, csandcf,

cs+cf =1. (7.1)

Applying the spatial averaging, we obtain from (6.2)–(6.3) (1−2cs) ˜gs f+Tln cs

1−cs =hs f. (7.2)

Here,

˜

gs f =gs f−gss/2;hs f =g⁰_f −g_s⁰−gss/2, (7.3) g_s⁰=X

k

c_k^∗g_k⁰+TX

k

c_k^∗lnc^∗_k, gss=X

gikc_i^∗c^∗_k, c^∗_i =ci/cs, (7.4)

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gss is the frustration parameter. It depends on the interaction coeﬃcients of thes-ﬂuctuons and probabilities©

c_i^∗ª

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

In the two-state approximation, the coeﬃcientgs f and the frustration parametergssare 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 externalﬁeldhs f. The solution of equation (7.2) atcs≪1is as follows:

cs(T)=exp©£

∆sf s

¡T_e⁰¢ ¡ T_e⁰−T¢

−gs f

¤/T_e⁰ª

. (7.5)

Here,

∆sf,s(T)= −

∂³ g⁰_s−g⁰_f´

∂T =sf(T)−ss(T) (7.6)

is the difference of entropies of thef- ands-ﬂuctuon.T_e⁰is the solution of the equation g⁰_f¡

P,T_e⁰¢

=g_s⁰¡ P,T_e⁰¢

. (7.7)

Atcf =1−cs≪1

cf(T)=exp©£

∆sf,s

¡T_e¹¢ ¡ T_e¹−T¢

−gs f

¤/T_e¹ª

, (7.8)

whereT_e¹is the solution of the equation g⁰_f ¡

P,T_e¹¢

=g_s⁰¡ P,T_e¹¢

+gss. (7.9)

The physical meaning of the characteristic temperaturesT_e⁰,T_e¹is explained below.

The temperatureTe, at which the“externalﬁeld”hs f is equal to zero, is the coexistence temperature of two heterophase liquid states determined by equation

g⁰_f (P,Te)=g_s⁰(P,Te)+gss/2. (7.10) AtT=Te, we havecs(Te)=cf(Te)=1/2. In the vicinity ofTe,

cs≈1

2+ hs f(T) 2(2Te−g˜s f)

"

1−

2T h²_{s f}(T) 3(2Te−g˜s f)³

#

=1

2+∆ss,f(Te) (T−Te) 2(2Te−g˜s f) +O¡

(T−Te)³¢

. (7.11)

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

T_e⁰≈Te+gss/2∆sf,s, T_e¹≈Te−gss/2∆sf,s. (7.12) The solution (7.11) is stable atg˜s f(P,Te)<2Te. Ifg˜s f(P,Te)>2Te, it is unstable and atT=Te, (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

1

e

Figure 1.(Color online) The solid-like fraction of of liquid,cs, vsT/T_e⁰at (a)g˜_{s f}(P,Te)>2Teand (b)

˜

g_{s f}(P,T_e)<2T_e.

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

Ifg˜s f =2Te, i.e.

gss=2gs f−4Te, (7.13)

then, the 2nd order phase transition takes place atT=Te(P). In accordance with (7.5) and (7.8), aboveT_e⁰ and belowT_e¹, the HPF are weak but within the temperature range£

T_e¹,T_e⁰¤

, where in compliance with (7.11)csandcf 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 vicinity of the phase coexistence temperatures (here,T_e⁰andT_e¹, respectively). Therefore,T_e⁰can be considered as the coexistence temperature of theﬂuid and heterophase liquid phases whileT_e¹is the phase coexistence temperature of the“ideal”glass (as it is determined above) and the heterophase liquid. Thus, T_e¹is the ideal glass transition temperature. The real glass transition temperature,Tg, which depends on τobs(see section 3), is aboveT_e¹due to dramatic retarding of the structure relaxation with temperature decrease. For this reason, the real glass transition temperature range,£

Tg,T_e⁰¤

, is narrower than£ T_e¹,T_e⁰¤

. The structure of the heterophase states in the vicinity of the characteristic temperaturesT_e¹,Te andT_e⁰ is schematically presented infigure 2. Infigure 3, the mesoscopic structure of the solid-like fraction with several types ofs-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 ofs-fluctuons{ci} ,i=1, . . . ,mare nearly constant or they are changing continuously and smoothly. This assumption fails if a phase transformation with stepwise changes of the fractions{ci}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 thes-ﬂuctuons is shown.

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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 thes-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:

¡1−2c₁^∗¢

csg˜12+Tln c₁^∗

1−c₁^∗=h12, c^∗_i =ci/cs, c₁^∗+c₂^∗=1 ,

g˜12=g12−¡

g12+g22¢

/2, h12=g₂⁰−g₁⁰+cs¡

g22−g11¢

/2. (7.14)

It is seen that this equation is isomorphic to equation (7.2) but the“externalfield”h12and the pair interaction coefficientcsg˜12 depend oncs. 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 arbitrary 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 reasonable specification of the coefficients.

As a useful example, let us consider solutions of equation (7.14) in the vicinity of the coexistence curve,h12(P,T)=0, assuming that cs¡

g22−g11¢

is a negligible quantity. In this case, the coexistence temperature,T12, is determined by equation

g₁⁰(P,T12)=g₂⁰(P,T12) . (7.15) It is assumed thatT12is above the coexistence temperatureTe. A phase transformation of the solid- like fraction and induced liquid-liquid phase transition atT12<Teis considered in [27].

In the vicinity ofT12

h12=g₁⁰(P,T)−g₂⁰(P,T)≈(s1−s2) (T−T12)≡∆s12(T−T12) , (7.16) s1ands2is the entropy ofs-ﬂuctuon of type 1 and 2, respectively.

If cs(T12) ˜g12>2T12, 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 parametercsatT=T12is [see (B.7) in appendix B] as follows:

∆cs(T12)≈(s1−s2)exp¡

−gs fβ¢

. (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).

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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 ofc₁^∗(T)shown in a) induces the appearance of the inﬂection point ofcs(T)atT=T₁₂, b).

If cs(T12) ˜g12<2T12, the fractionsc₁^∗,c^∗₂ and parametercs change continuously atT =T12but the inﬂection points of the functionsc₁^∗(T)and cs(T)appear atT =T12 (ﬁgure 5). Hence, the expression (7.17) estimates the bench height of the parametercs.

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

∆H12≃cs(T12) (s1−s2)k₀⁻¹£

Hf s+T12

¤, (7.18)

Hf 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(s1−s2)k₀⁻¹T12.

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

Atcsg˜12<0, the“antiferromagnetic”order with alternatings-ﬂuctuons of different types is preferable.

In both cases, theﬂuctuonic SRO generates molecular medium-range order with the correlation length ξf l∼2r0[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,ξFC≫ξf l. It appears due to the aggregation of thes-fluctuons under the effect of the volumetric gravitation potential (4.6). The equilibrated aggregation ofs-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 atT =293K, while theα-relaxation time at this temperature isτα=40ns. With temperature increase, the speckle size and contrast decreases and atT >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,τFC, is shorter than the observation time, i.e., ifτα≪τFC≪τobs≪τLRO. The heterogeneous structure and slow structural relaxation are observed not only in many Van der Waals molecular liquids but also in some metallic melts aboveTm[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

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

proportional to the isothermal compressibilityκT and are independent of the wave vectorq,

¯

¯ρ(q)¯

¯

2|q→0∼κTT. (7.19)

Here,qis 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 aq-dependent excess scattering intensity,Iexc

¡q¢

∼q⁻^D (Dis the fractal dimension) occurs atT<TA≈T_e⁰. TheIexc

¡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 atT <TA[9, 61]. It turns out that the thermodynamics andα-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 lengthξf l≈2r0, (section 4).

Theory of the Fischer cluster is developed in [10, 23–26]. The Fischer cluster is considered as a fractal aggregation ofs-ﬂuctuons with the fractal dimensionDf and correlation lengthξFC. Minimization of the free energy as function ofc¯s,Df andξFCallows one to determine the equilibrium values ofDf andξFC. It is found (see appendix C) that

ξFC

¡cs,Df

¢≈¡ cs

¢

1

D f−3r0, Df =3−

µ ln r0

ξf l

¶₋1

lncs,CD, (7.20)

cs,CDis the concentration ofs-ﬂuctuons within CD ([24–26], appendix C)

cs,CD=

"

1+ I(Df,ξFC)

∂²g⁰¡ cs

¢/∂c²_s−I(Df,ξFC)

#

csÊcs. (7.21)

The quantityI(Df,ξ), (C.21), is proportional toφ0. Equation (7.21) shows that within the CD, concentration ofs-ﬂuctuons is larger than its mean valuecs:cs,CD≃cs+^constφ0.

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

cs>cs,0<

µr0

ξf l

¶2

≈0.16, 1<Df 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∝ϕ0.

The upper bound of thecs-range in which the Fischer cluster exists,cs,1=(ξf l/r0)²cs,0 [see (C.26)], decreases,∼φ0, with an increase of the strength of thes-ﬂuctuons gravitation potentialφ0. When cs

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Figure 7.(Color online) Parametric phase diagram on the plane¡ T^∗,g^∗_ss¢

,T^∗=T/T_e⁰,g_ss^∗=gss/∆s_s,fT_e⁰. The phase coexistence linesT=T_e⁰;T=Te;T=T_e¹and the threshold of the Fischer cluster formation temperature,T_FC^∗, approximately determined using equations (7.5), (7.22) are shown. The critical end point atg_ss,^∗_cexists on the lineT=Te. Bold line 1 schematically presents an evolution phase curve of the equilibrium system.

approachescs,1<1from below, the fractal dimensionDf approaches 3 andξFC→ ∞. It means that at csÊcs,1the solid-like fraction consists of the connected 3-dimensional solid-like clusters of sizer>ξf l. Thus, atcsÊcs,1, the topology of the heterophase liquid equilibrated on scalerÊξf lchanges.

7.4. Parametric phase diagram

In the HPFM, the structure and phase states of the heterophase liquid are described in terms ofT and coeﬃcientsgs f,gss,hs f. It is useful to construct a phase diagram (the parametric phase diagram) of the glass-forming liquid in terms of these parameters⁵. 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 gs f,gss,hs f. The quantitiesT_e⁰,T_e¹,Te andTFCare coexistence temperatures of

1) theﬂuid and heterophase liquid (T_e⁰);

2) the heterophase liquid and“ideal”glass(T_e¹);

3) theﬂuid-like and solid-like states (Te);

4) the heterophase liquid with and without the Fischer cluster (TFC).

Introducing the scaled temperature,T^∗=T/T_e⁰, and the frustration parameterg_ss^∗=gss/∆sf,sT_e⁰, we can present the relations (7.8) in a dimensionless form,

T_e⁰^∗=1, T_e^∗=1−g_ss^∗/2, T_e¹^∗≈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=2gs f

± ¡∆ss,fT_e⁰¢

−4Te^∗(P)>0. (7.24)

Theﬁrst orderﬂuid-solid phase transition on the phase coexistence curve takes place atg_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 temperatureT_g^∗. Within the range0<g_ss^∗<g^∗_ss,_cthefirst order phase transition takes place on the phase coexistence lineT^∗=T_e^∗. A weakfirst order phase transition takes place on the lineT^∗=T_FC^∗.

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

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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⁻^Datr0≪q⁻¹≪ξ⁻¹. Atqr0∼1, it is a superposition of the pair correlation functions ofﬂuctuons,

⌢̟(q,T)=cf(T)̟f(q)+

m

X

i=1

ci(T)̟i(q)=[1−cs(T)]̟f(q)+cs(T)̟s(q), (7.25)

̟s(q)= 1 cs

m

X

i=1

ci̟i(q), (7.26)

̟f(q),̟i(q)are Fourier transforms of the pair correlation functions of the f- ands-ﬂuctuons, respectively. The cross-correlation terms∼cfcs are omitted in (7.25). The quantities̟f(q),̟i(q)weakly depend on the temperature. For this reason, the equation (7.25) can be applied to restore the order parametercs(T)using the structure factors̟f(q),̟s(q)measured in the liquid,ﬂuid and glassy states [61]. On the other hand, as it is shown in [22],cs(T)can be restored from calorimetric data using the relation

cs≃£

Hf(T)−Hexp(T)¤ /£

Hf(T)−Hs(T)¤

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

T_e¹,T_e⁰¤

, andHexp(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 parameterg˜s f are used, is presented there by a solid line.

Let us remind that the analytic solution describes the order parametercs(T)of the equilibrated system. Therefore, it noticeably deviates from the experimentally determined valuescs(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 ofcs.

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

Arrows indicate the temperaturesT_e⁰,Te,Tg,T_e¹. Line presents an analytic solution of the equation of state in the two-state approximation withﬁtting parameterg˜_{s f}.

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8. Dynamics

8.1. α -relaxation

Thermally activated cooperative structural rearrangements which can involve up to∼10²molecules [19, 20, 63–67] are calledα-relaxation. A large amount of the molecules are involved in the rearrangement due to correlations. Structural rearrangement of aﬂuctuon also involves rearrangements of the neighboringﬂuctuons within CD of sizeξf l. Therefore, the size of cooperatively rearranging domain is nearly equal toξf l.

The activation energy ofα-relaxation,

Eac=dlnτα

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,

Eac=E⁰_ac+E_ac¹cs+E_ac²c²_s+. . . . (8.2) AboveTA, the activation energy is nearly equal toE⁰_ac. Cooperativity of the liquid dynamics is induced by thes-ﬂuctuons interaction which becomes considerable belowTA.

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 scaleξf l. This assumption leads to the following expression [22]

Eac= A

(1−TK/T)²+zCDcs

¡Hf −Hs

¢+O¡ c²_s¢

. (8.3)

zCD∼¡ ξf l/a¢3

is the cooperativity parameter, which is the mean n umber of molecules within the CD;Hf, Hsis 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 aboveTA. The Kauzmann temperature,TK, is afitting parameter (see comments concerningTKin section 3)

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

EnthalpiesHf(T)andHs(T)within the temperature range£ Tg,TA¤

are understood as extrapolations of these functions measured atT >TAandT<Tg, 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 parametersA=967K;TK=153K,

6There is no reason to believe thatEac(cs)is a singular function atcs∈[0,1].