Heterophase liquid states: Thermodynamics, structure, dynamics

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 heterophase fluctuations 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.


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 amorphous (glassy) solid only if cooling is fast enough to avoid crystallization. As a result, the liquid is nonequilibrium 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.
The HPFM is based on the statistics of the transient solid-like and fluid-like mesoscopic species (clusters) which are called s-fluctuons and f -fluctuons, respectively. By definition, each fluctuon is specified by SRO. The minimal size of a fluctuon is equal to the SRO correlation length, ξ SRO . An arbitrary number of types of the s-fluctuons, m 1, 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 per fluctuon equal to k 0 ∼ r 3 0 . 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 and fluid-like fractions consist of s-and f -fluctuons, respectively. Let us denote by N the total number of molecules of liquid and byN f , N 1 , . . . , N m the numbers of molecules belonging to f -and s-fluctuons,

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, τ obs , should be less than the time of crystallization, τ obs ≪ τ LRO , (3.1) τ LRO is 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 elementary 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 obs from above. The temperature-time-transformation diagram can be used to estimate τ LRO and 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 obs from 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 .

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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 g determined 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 satisfied, 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 noncorrelated and statistically independent if the distance between them exceeds the largest correlation length which is finite 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 configurational entropy, S conf (T ), as a function of temperature, at a finite temperature T K (the Kauzmann paradox) [32], and Vogel-Fulcher-Tamman singularity of τ α (T ) at a temperature T VFT [33][34][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 VFT found from the fittings 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 K and T VFT should be considered as free parameters of the widely used phenomenological model [36]. The issue of proximity of T K and T VFT is considered and confirmed within the framework of HPFM in [37].

Mesoscopic free energy of the heterophase liquid
The phenomenologic free energy of the heterophase liquid in terms of the introduced order parame-  In the summand G L (P, T ), just local interactions of the fields {c i (x)} are included, g 0 i (P, T ) is independent of the order parameter free energy of i -th fluctuon; g 0 ik (P, T ) 0 is the fluctuonic pair interfacial free energy; z is the fluctuonic 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 (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 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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Heterophase liquid states the s-fluctuons, leading to the Fischer cluster formation. It is taken as Yukawa potential with cutoff range R 0 which is larger than but comparable with r 0 , Fluctuonic short-range correlation appears due to both local and volumetric interactions. The Ornstein-Zernike (OZ) equation [38] can be used to estimate the fluctuonic 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 correlation function, which, in turn, is comparable with the range of the fluctuonic pair interaction potential.
With R 0 2r 0 we have that ξ f l ≃ 2r 0 ≃ 2ξ SRO . As it is seen, the ordering of fluctuons causes extension of the molecular pair correlations beyond r 0 and the formation of the of molecular MRO. The liquid region of size ξ f l with correlated fluctuons is referred to as correlated domain. The fact that the components of the order parameter A i (x) are normalized probabilities, which cannot exceed 1, validates the presentation of G (P, T ) in the form of the polynomial expansion in powers of 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.

The fluctuon-fluctuon interaction and the frustration parameter
The physical meaning of the pair interaction coefficients of the neighboring fluid-like and solid-like fluctuons is clear. It is the fluid-solid interfacial free energy taking into account the geometry of the contacting fluctuons.
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, coherent joints of the non-crystalline s-fluctuons is hampered at any orientation. The interfacial free energy increase due to the geometric badness of the fit of contacting s-fluctuons is the structural frustration parameter 4 . Because of its importance, let us consider the fluctuonic 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 a z-vertex coordination polyhedron in the case when the addition of a new coordination shell leads to the formation of z − 1 new 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 + 1 molecules is E z+1 = ε 0 (z + 1) +ε def (z − 1) + ε 1 = ε 0 (z + 1) + ε frust , (5.1) ε 0 is the mean energy of the initial cluster,ε def is the mean energy of deformation and ε 1 is the energy of a molecule with the coordination polyhedron of different geometry. The last two terms in r.h.s. of 4 For 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, 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: As it is seen, ε frust is ∼ z while s frust ∼ ln z. Therefore, g frust > 0 with z ≫ 1.
One can conclude that generally the structure of interfacial layer of contacting fluctuons is frustrated and that g frust > 0.

Equations of the liquid state equilibrium
Variation of the free energy functional (3.1) at condition (2.3) yields the equations of equilibrium state, λ is the Lagrange multiplier. Let us denote by µ i (P, T ) the derivative Here, g ik = zg 0 ik . Variables (P, T ) are not shown.
As a result, it follows from (6.1) that Equilibrium state is stable if the quadratic form
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.
Applying the spatial averaging, we obtain from (6.2)-(6.3) Here,g (7.4) g ss is the frustration parameter. It depends on the interaction coefficients of the s-fluctuons and probabilities c * i . 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 constants. 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 external
field h s f . The solution of equation (7.2) at c s ≪ 1 is as follows: Here, is the difference of entropies of the f -and s-fluctuon. T 0 e is the solution of the equation where T 1 e is the solution of the equation The physical meaning of the characteristic temperatures T 0 e , T 1 e is explained below.
The temperature T e , at which the "external field" h s f is equal to zero, is the coexistence temperature of two heterophase liquid states determined by equation As it follows from (7.7), (7.9) and (7.10), T 0 e ≈ T e + g ss /2∆s f ,s , T 1 e ≈ T e − g ss /2∆s f ,s .
The solution (7.11) is stable atg s f (P, T e ) < 2T e . Ifg s f (P, T e ) > 2T e , it is unstable and at T = T e , (P ) the first order phase transition takes place.  Graphic representation of solutions of equation (7.2) is shown in figure 1. The stable and unstable solutions are depicted by solid and dashed lines, respectively. (7.13) then, the 2nd order phase transition takes place at T = T e (P ). In accordance with (7.5) and (7.

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: 14) It is seen that this equation is isomorphic to equation (7.2) but the "external field" h 12 and the pair interaction coefficient c sg12 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 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, 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 It is assumed that T 12 is above the coexistence temperature T e . A phase transformation of the solidlike fraction and induced liquid-liquid phase transition at T 12 < T e is considered in [27].
In the vicinity of T 12

The Fischer cluster
Along with the above considered types of MRO appearing due to local fluctuonic interaction, there is a different type of the fluctuonic order with comparatively large (as it was observed, up to ∼ 102 nm) 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 depending on the liquid features and temperature. This remarkable phenomenon, which was discovered and investigated in detail by Fischer et al. [1][2][3][4][5][6][7][8][9][10][23][24][25][26][27], is known as the Fischer cluster. The Fischer cluster was visualized by observing a speckle pattern in ortho-terphenil [6]. The speckle pattern fluctuates and rearranges very slowly, with characteristic time ∼ 1 min at T = 293 K, while the α-relaxation time at this temperature is τ α = 40 ns. With temperature increase, the speckle size and contrast decreases and at T > 340 K no speckle is seen. Schematically, the heterophase liquid structure with and without the Fischer cluster is shown in figure 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 relaxation 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 density fluctuations. The conventional large-scale density fluctuations in a homophase liquid are 43701-10 proportional to the isothermal compressibility κ T and are independent of the wave vector q, Here, q is the wave vector, ρ(q) is the amplitude of density fluctuations. The intensity of X-ray scattering on the density fluctuations, 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 0 e . The I exc q is much larger than the scattering intensity on the thermal fluctuations

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 Introducing the scaled temperature, T * = T /T 0 e , and the frustration parameter g * ss = g ss /∆s f ,s T 0 e , we can present the relations (7.8) in a dimensionless form, (7.23) The end critical point location on the fluid-solid phase coexistence curve, T * = T * e (P ) is located at The first order fluid-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 in figure 7. As an example, here is also shown one phase trajectory which becomes nonphysical below the glass transition temperature T * g . Within the range 0 < g * ss < g *

Static structure factor and the order parameter restoration
Pair correlation function of the density fluctuations , ⌢ ̟ (q, T ), of the heterophase liquid with the Fischer cluster is ∼ q −D at r 0 ≪ q −1 ≪ ξ −1 . At qr 0 ∼ 1, it is a superposition of the pair correlation functions of fluctuons, 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 in figure 8 by scattered symbols. Solution of the equation of state in the twostate 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 parameter c s (T ) of the equilibrated system. 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"

α-relaxation
Thermally activated cooperative structural rearrangements which can involve up to ∼ 10 2 molecules [19,20,[63][64][65][66][67] are called α-relaxation. A large amount of the molecules are involved in the rearrangement due to correlations. Structural rearrangement of a fluctuon also involves rearrangements of the neighboring fluctuons 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, depends on the order parameter. It can be presented as an expansion in powers of the order parameter [10,22] 6 , Fischer and Bakai [22] have suggested that CD can be rearranged when all the molecules therein are in fluid-like state with correlations destroyed on the scale ξ f l . This assumption leads to the following expression [22] E ac = A 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. The first 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 the fluid and the numerator is equal to the activation energy above T A . The Kauzmann temperature, T K , is a fitting parameter (see comments concerning T K in 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 in figure 9. The experimental data are shown by circles. The curve is a result of fitting the formula (8.3) using parameters A = 967 K; T K = 153 K, T e = 257 K; T A ≈ 325 K; z CD = 32; k 0 = 7. The input of the first term of r.h.s of (8.3) in the activation energy is nearly 10% at T = T g .
It is noteworthy that since the main input in the activation energy makes the term proportional to c s , E ac (T ) has the inflection point at T ≈ T e . Stickel has proposed an efficient method of analysing the α-dynamics to check the applicability of the Vogel-Fulcher-Tamman formula and other phenomenologic and empiric expressions proposed for τ α (T ) [70][71][72]. He has analyzed many molecular liquids and found the flex points of E ac (T ) identified as T e . Coincidence of the values of T A and T e extracted from the dynamic, calorimetric and structural data (figures 8, 9) support the adequacy of HPFM.

Ultra-slow modes and the Fischer cluster equilibration time
Two relaxational modes are connected with the Fischer cluster. The phase transformation of a liquid without the Fischer cluster into the state with the cluster is controlled by nucleation and growth of a new phase. Since the phase transformation heat is small, the phase equilibration time is rather large compared with the time of elementary cooperative rearrangement τ α . The Fischer cluster equilibration time is determined in [10], Rearrangements of the equilibrated Fischer cluster on the scales ξ −1 f l ≫ q > ξ −1 FC are registered as the ultraslow modes [2][3][4][5][6][7][8][9]. The relaxational rate of the ultra-slow mode is ∼ q 2 . It is found within the framework of HPFM [10] that The characteristic times Γ −1 usv and τ FC are proportional to τ α and the proportionality coefficients are rather large at ξ FC ≫ ξ f l . As it is seen, τ FC ≫ Γ −1 usv ≫ τ α . Relations (8.4), (8.5) are in harmony with the experimental data.

Fragility
The fragility parameter, ⌢ m, introduced by Angell in [73,74], is an important characteristic of the glassforming liquid dynamics near T g . It is taken as the measure of deviation of the temperature dependence of τ α from the Arrhenius law. There exist strong liquids, with small fragility parameter, ⌢ m ∼ 10, the most fragile liquids, with ⌢ m ≈ 10 2 , and liquids with moderate fragility in between. The fragility parameter is tightly connected with the structural properties and thermodynamics of a liquid, and for this reason, it is widely used at analysing the glass transition and classification of liquids. Angell's definition of this parameter is as follows: In HPFM, the quantity E ac is determined by equation (8.3). As it follows from (8.3) and (8.6), The main contribution in ⌢ m gives the second term within the brackets of (8.7). It is proportional to the number of molecules involved in the cooperative rearrangement, z CD ∼ ξ 3 f l , as well as to the difference of the configurational entropies of the solid-like and fluid-like species. The difference of their vibrational entropies is comparatively small. Since these quantities can be measured regardless of τ α , equation (8.7) permits to check the relevance of the HPFM predictions. For example, it was found that for salol, HPFM gives ⌢ m ≈ 67 [10]. The fragility parameter of salol estimated in [75] is equal to 63.

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As it was noted above, (subsection 8.3), the 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. Naturally, the linear or quadratic extrapolation provides an acceptable result if the function is smooth and the higher derivatives are small. Phase transformations in the solid-like fraction lead to stepwise changes of H s (T )and, consequently, to the stepwise behavior of τ α (T ). In this case, extrapolations of τ α (T ), determined by equation (8.1), from high and low temperatures into the range T g , T A cannot be properly fitted. Equations (7.17), (7.18), (8.1)-(8.3) determine the temperature dependence of τ α (T ) in this case.
In a series of experiments with some metallic glasses [76][77][78][79][80], the fragility parameter value determined using the data on τ α (T ) in the vicinity of T g and its value recovered from the extrapolated curve τ α (T ) measured at high temperatures are completely different. As it is revealed [76], such a behavior of τ α (T ) of Zr-based alloy Vitreloy 4 is connected with the liquid-liquid first order phase transition. In others melts, a transition of this type is assumed. Equations (7.17), (7.18), (8.1)-(8.3) provide theoretical description of this phenomenon known as the fragile-to-strong liquid transformation. More in detail it is considered in [81]. ξ f l and τ α , respectively. Therefore, the formation of the crystalline embryos with the size larger than ξ f l takes much longer time than the SRO equilibration time. As a result, the condition (3.2) can be regarded as satisfied when the condition (3.1) is fulfilled. Hence, the crystalline species of size ∼ ξ f l coexist with non-crystalline species within the solid-like fraction of liquid and in glass. As a confirmation, direct observations of the structure of metallic glasses by means of a high resolution field ion microscopy and transmission electron microscopy (see figure 6.5 in [21], [82,83] and references cited) reveal the coexistence of crystalline and non-crystalline structural species with sizes of up to a few nanometers. The Fischer cluster equilibration (along with the ultra-slow modes) is observable only if the crystallization time is much longer than τ (ξ FC ). The crystallization heat (which is the thermodynamic driving force of the crystallization) is much larger than the heat of the Fischer cluster formation. Therefore, the Fischer cluster can be observed just in normal liquids and in the supercooled liquids with strongly hindered crystallization.

Concluding remarks
The amount of the solid-like fraction, c s , determines the measure of the fluid-to solid transformation. Due to the definitive role of c s (T ), its description is an important issue of the theory. The two-state approximation is a minimal model permitting to solve this problem considering the fluid-solid HPF states without details of the solid-like subsystem. Evidently, this model is satisfactory if just one type of the sfluctuons is statistically significant or when variations of the probabilities c * i within the glass-transition temperature range are insignificant, i.e., if the mesoscopic structure of the solid-like fraction does not vary considerably. At the same time, estimation of the two-state approximation accuracy shows that it can yield acceptable results in more general cases.
The accuracy of the two-state approximation can be estimated considering the states with transforming s-fluctuons. The assumption on a smooth evolution of the coefficients of equation (7.2) fails if the

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phase transformations, similar to those considered in subsection 7.2, take place within the solid-like fraction. Nevertheless, even in this case, a stepwise jump of c s (T ) is comparatively small because the entropy jump and transformation heat at the solid-solid polymorphic transformation, as a rule, is small, ∆s 12 ∼ 10 −1 while ∆s f s ∼ 1. Therefore, one can expect that the two-state approximation is acceptable with accuracy to terms O 10 −1 c s or even better.
Figuratively speaking, in a general case, the glass and solid-like fraction of a liquid is a mosaic composed by the mesoscopic species of size ∼ r 0 ( figure 3). However, unless until the mutual ordering of s-fluctuons and the impact of the mosaic details on c s is beyond the scope of interests, the two-state approximation can be used. The question whether glass is a non-equilibrated highly viscous liquid or it is a non-equilibrated solid has rather got a conceptual sense. The thermodynamic continuity of the glass transition permits to believe that glass is a liquid with very high viscosity and long equilibration time. But, as a matter of fact, the glass near and below T g , with c s → 1, is solid with statictically insignificant amount of the fluid-like species.
Nevertheless, it flows, like a polycrystal does, due to the diffusional-viscous flow [84]. Field-emission microscopy of metallic glasses visualizes their grainy (polycluster) structure with sizes of grains ∼ 102 nm. The Coble mechanism of the plastic deformation [85,86] prevails near T g in such a glass [84]. The grainy structure of glass is the result of the existence of many centers of solidification within the liquid. Therefore, the polycluster mosaic structure of glass forms in liquids with different features of molecular forces. Slow structural relaxation hinders the "reclusterization" processes and the formation of "ideal" glass.
The Fischer cluster topology changes with an increase of the solid-like fraction. Its fractal dimension D f is less than 3 at c s < c s,1 and it is equal to 3 at c s c s,1 (see subsection 7.3 and appendix C). Thus, at the point c s = c s,1 = 1 − const φ 0 , the topological transition takes place at which the heterophase correlated domains transform into homophase ones. It is important that this transition does not presuppose the Fischer cluster equilibration on scales ξ FC > ξ f l . This result denotes a change of the structural relaxation mode at glass transition considered in [84]. The mesoscopic theory of thermodynamics and dynamics of the glass-forming liquids and glasses is connected with the microscopic approach based on the consideration of the potential energy landscape (see [87] and references cited) by the landscape coarsening procedure used while deducing the efficient Hamiltonian (appendix A). The coefficients of the fluctuon interaction save the memory on the microscopic potential energy landscape.

A. The bounded phase space and efficient mesoscopic Hamiltonian
Below T m , a crystalline state is the most probable one. It occupies a phase space region Ω cr of the total phase space Ω. Ω can be presented as the sum of the regions belonging to crystalline and non-crystalline states, Ω = Ω cr + Ω ncr .
Excluding Ω cr , we obtain the bounded phase space belonging to non-crystalline states. The bounded partition function determines the free energy of the non-crystalline state In [39], the procedure of derivation of the equation of the free energy in terms of the order parameter

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Here, c i q; x, p are Fourier transforms of the components of the order parameter at a fixed coordinate (x, p) of the bounded 6N-dimensional phase space Ω ncr .
Performing the integration in the functional space, we have G (P, T ) = −T ln exp −G c( q) β q,i dc i q . The termG(P, T ) takes into account the spatial fluctuations of the order parameter with qr 0 ∼ 1. To include this summand into consideration is important in the vicinity of critical points. It generates random fields and has an impact on criticality. It is shown in [10] that at the end point on the phase coexistence curve, the first order phase transition can take place due to the impact of the random field.
Within the framework of the method of cooperative variables, used while considering the gas-liquid critical point [88] and systems with the Ising-type Hamiltonian [89], a procedure of accounting ofG(P, T ) in the vicinity of the critical point is expounded.

B. Solutions of the equations of state
To get solutions of the equations (7.1)-(7.4), (7.14), (7.16), let us consider the solutions of equations (7.14), (7.16) taking c s (T ) as an unknown function which should be determined later on using equations (7.2)-(7.4).