Loosing thermodynamic stability in amorphous materials

The primary relaxation dynamics near the glass transformation temperature T g exhibits universal features in all glass formers, when showing two-level tunneling states (Low Temp. Phys. 35, 282 (2009)). Researchers have long searched for any signature of the underlying “true” ergodic–nonergodic trans...

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Datum:	2011
1. Verfasser:	Kokshenev, V.B.
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Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2011
Schriftenreihe:	Физика низких температур
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spelling	irk-123456789-1185502017-05-31T03:03:10Z Loosing thermodynamic stability in amorphous materials Kokshenev, V.B. 8th International Conference on Cryocrystals and Quantum Crystals The primary relaxation dynamics near the glass transformation temperature T g exhibits universal features in all glass formers, when showing two-level tunneling states (Low Temp. Phys. 35, 282 (2009)). Researchers have long searched for any signature of the underlying “true” ergodic–nonergodic transition emerging at a certain thermodynamic instability temperature Te . Here, the relaxation timescale for glass-forming materials is analyzed within a self-consistent thermodynamic cluster description combined with the cluster percolation concept. Exploring the ergodic hypothesis, its violation is found near a crossover from the Gaussian to non-Gaussian (Poisson) cluster-volume fluctuations, describing the finite-size fractal-cluster distributions. The transformation of the compact-structure “ergodic” clusters into hole-like glassy nanoclusters is attributed to the critical-size thermal fluctuations. The ergodic–nonergodic phase diagram showing Te is predicted in the model-independent form through the glass fragility parameter known for organic and inorganic liquids and amorphous solids. In all cases the ergodic-instability temperature is located below and close to the glass transformation temperature, whereas the distance between the two characteristic temperatures decreases with growing the material fragility. 2011 Article Loosing thermodynamic stability in amorphous materials / V.B. Kokshenev // Физика низких температур. — 2011. — Т. 37, № 5. — С. 551–557. — Бібліогр.: 26 назв. — англ. 0132-6414 PACS: 61.41.+e, 61.43.Fs, 64.70.P– http://dspace.nbuv.gov.ua/handle/123456789/118550 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	8th International Conference on Cryocrystals and Quantum Crystals 8th International Conference on Cryocrystals and Quantum Crystals
spellingShingle	8th International Conference on Cryocrystals and Quantum Crystals 8th International Conference on Cryocrystals and Quantum Crystals Kokshenev, V.B. Loosing thermodynamic stability in amorphous materials Физика низких температур
description	The primary relaxation dynamics near the glass transformation temperature T g exhibits universal features in all glass formers, when showing two-level tunneling states (Low Temp. Phys. 35, 282 (2009)). Researchers have long searched for any signature of the underlying “true” ergodic–nonergodic transition emerging at a certain thermodynamic instability temperature Te . Here, the relaxation timescale for glass-forming materials is analyzed within a self-consistent thermodynamic cluster description combined with the cluster percolation concept. Exploring the ergodic hypothesis, its violation is found near a crossover from the Gaussian to non-Gaussian (Poisson) cluster-volume fluctuations, describing the finite-size fractal-cluster distributions. The transformation of the compact-structure “ergodic” clusters into hole-like glassy nanoclusters is attributed to the critical-size thermal fluctuations. The ergodic–nonergodic phase diagram showing Te is predicted in the model-independent form through the glass fragility parameter known for organic and inorganic liquids and amorphous solids. In all cases the ergodic-instability temperature is located below and close to the glass transformation temperature, whereas the distance between the two characteristic temperatures decreases with growing the material fragility.
format	Article
author	Kokshenev, V.B.
author_facet	Kokshenev, V.B.
author_sort	Kokshenev, V.B.
title	Loosing thermodynamic stability in amorphous materials
title_short	Loosing thermodynamic stability in amorphous materials
title_full	Loosing thermodynamic stability in amorphous materials
title_fullStr	Loosing thermodynamic stability in amorphous materials
title_full_unstemmed	Loosing thermodynamic stability in amorphous materials
title_sort	loosing thermodynamic stability in amorphous materials
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2011
topic_facet	8th International Conference on Cryocrystals and Quantum Crystals
url	http://dspace.nbuv.gov.ua/handle/123456789/118550
citation_txt	Loosing thermodynamic stability in amorphous materials / V.B. Kokshenev // Физика низких температур. — 2011. — Т. 37, № 5. — С. 551–557. — Бібліогр.: 26 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT kokshenevvb loosingthermodynamicstabilityinamorphousmaterials
first_indexed	2025-07-08T14:13:30Z
last_indexed	2025-07-08T14:13:30Z
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fulltext	© Valery B. Kokshenev, 2011 Fizika Nizkikh Temperatur, 2011, v. 37, No. 5, p. 551–557 Loosing thermodynamic stability in amorphous materials Valery B. Kokshenev Departamento de Física, Universidade Federal de Minas Gerais, Instituto de Ciências Exatas Caixa Postal 702, CEP 30123-970, Belo Horizonte, Brazil E-mail: valery@fisica.ufmg.br Received December 1, 2010 The primary relaxation dynamics near the glass transformation temperature gT exhibits universal features in all glass formers, when showing two-level tunneling states (Low Temp. Phys. 35, 282 (2009)). Researchers have long searched for any signature of the underlying “true” ergodic–nonergodic transition emerging at a certain thermodynamic instability temperature eT . Here, the relaxation timescale for glass-forming materials is ana- lyzed within a self-consistent thermodynamic cluster description combined with the cluster percolation concept. Exploring the ergodic hypothesis, its violation is found near a crossover from the Gaussian to non-Gaussian (Poisson) cluster-volume fluctuations, describing the finite-size fractal-cluster distributions. The transformation of the compact-structure “ergodic” clusters into hole-like glassy nanoclusters is attributed to the critical-size thermal fluctuations. The ergodic–nonergodic phase diagram showing eT is predicted in the model-independent form through the glass fragility parameter known for organic and inorganic liquids and amorphous solids. In all cases the ergodic-instability temperature is located below and close to the glass transformation temperature, whe- reas the distance between the two characteristic temperatures decreases with growing the material fragility. PACS: 61.41.+e Polymers, elastomers, and plastics; 61.43.Fs Glasses; 64.70.P– Glass transitions of specific systems. Keywords: glass forming materials, ergodic hypothesis, thermodynamic instability. 1. Introduction The structural transformation, which occurs under cool- ing rates preventing formation of the long-range crystalline order, is essentially a crossover from the high-temperature thermally equilibrated ergodic state to low-temperature nonergodic glassy states, characteristic of amorphous sol- ids studied near the glass transformation temperature gT via scanning calorimetry, e.g., [1–3]. Researchers have long searched for a signature of the underlying “true” ergo- dic–nonergodic transition emerging at a certain ergodic- instability temperature, designated by the critical tempera- ture at which the certain physical characteristics exposes a divergent behavior. A traditional approach is mapping the order–disorder thermodynamic transitions onto the geome- tric, cluster percolation picture, e.g., [4]. There are several ways of description of the nontrivial ergodicity breaking in spin-glass theories. One example is the incorporation of the cluster-distribution function into a percolation free-energy by means of employing of the ana- logy with the Ising model [5] where the cluster distribution function exhibits a singularity near the percolation thre- shold [4]. It has been also recognized that in the formation of the collective order specific of metastable glassy-like states, the singularities in the cluster-size distributions are avoided, by both the asymptotically small and large clus- ters. It has been shown in [6] for the specific case of super- cooled liquids (SCLs) that the ergodicity breaking can be developed through a dynamic crossover from the ideal-gas state to nonideal gas, characterized by the strengthening of the intermolecular correlations. Such a molecule- correlation crossover in the ergodic gas system was illumi- nated through the description of a smooth transformation from the Gaussian-type to Poisson-like volume-molecule fluctuations. The Gaussian-to-Poisson crossover in cluster distribution was first observed by Chamberlin et al. [7] in a number of SCLs, below the conventional glass transition temperature gT through the stress relaxation data. A corre- lation between nonergodicity and non-Gaussianity in the glass formation process was also later discussed by Oda- gaki [8] and experimentally tested by Colby [9]. In the present study, the theoretical approach to the problem of the ergodic–nonergodic instability communi- cated in [6], is improved and extended over inorganic li- Valery B. Kokshenev 552 Fizika Nizkikh Temperatur, 2011, v. 37, No. 5 quids and amorphous solids. Equation for eT is re-ana- lyzed and presented in the model-independent form that provided the ergodic instability boundary in different glass forming materials. This boundary now is tested by recent experimental data obtained for glass-forming polymers and metal alloys. 2. Background 2.1. Phenomenological and model forms The phenomenological Vogel–Fulcher–Tammann (VFT) fitting form, namely ( ) ( ) min 0 = exp , with = 1VFT VFT TT T D T T ⎛ ⎞ τ τ ε −⎜ ⎟ε⎝ ⎠ , (1) which also reads as ( ) ( ) 0 10 10 min 0 = , with = ,log log ln10 VFT VFT T DTB B T T τ τ + − (2) is widely used to describe the non-Arrhenius temperature behavior of the structural relaxation times observed in amorphous liquids and solids; D is the so-called strength index [10,11] and 0T is the VFT temperature. The VFT form performs within the temperature domain [12] <g cT T T≤ , where cT is the crossover temperature cT between the moderately and strongly supercooled liquid states [12], distinguished in the mode coupling theory [1]. In order to characterize the timescale temperature beha- vior, the timescale steepness function 10log ln = ln ln10 T T T d dTm d T dT τ τ ≡ − − (3) is also defined [6,12]. When applied to Eq. (1) at = gT T , one arrives at the glass-former fragility [11] 0 1= 1 , with = 1g g g g g T m m T ∗ ⎛ ⎞ + ε −⎜ ⎟⎜ ⎟ε⎝ ⎠ , (4) where ( )= VFT g gm m∗ ∗ is the material-independent lower limit, following from Eq. (1), that can be also presented in the mo- del-independent form (exp) (exp) 10= ( / )logg gm∗ ∞τ τ . The inser- tion of the estimates (exp) 2 1= 10 sg ±τ and (exp) 14 2= 10 s− ± ∞τ in Eq. (4) and gm∗ yields the well- known constraint for the characteristic-temperature ratio [13,11] 0 = , = 16 2.g g g g g T m m T m m ∗ ∗ ± − (5) The experimental validity of Eq. (5) for SCLs was tested in Fig. 2 in Ref. 12, along with similar equation = , = 7 1,g cc c g g c m mT m T m m ∗ ∗ ∗ + ± − (6) obtained [14] for the the crossover temperature, where cm∗ is the corresponding lower fragility limit. In the seminal thermodynamic model by Adam and Gibss (AG) [15], the dynamic properties of SCLs are de- scribed by ( ) ( ) ( ) min= exp AG AG AG T T B n k T ⎛ ⎞Δμ τ τ ⎜ ⎟⎜ ⎟ ⎝ ⎠ (7) obtained through the average transition probability ( )1/ AG Tτ characteristic of the smallest-size cooperatively rearranging regions (CRRs). Here ( )AGΔμ is the molar (so- lid-over-liquid excess) chemical potential, approximated by a constant, whereas Tn stands for the mean number of molecules which constitute the rearranging region. The SCLs were studied [17] simultaneously on the ba- sis of the dynamic data (exp) Tτ , derived from the dielectric loss spectra, and the thermodynamical experimental data on the configurational entropy, namely (liq) (sol)= , = , T T T T T T TK C S dT C C C T Δ Δ Δ −∫ (8) evaluated through the excess liquid-over-solid isobaric specific heat TCΔ . The thermodynamic Kauzmann tempe- rature [18] KT is defined by the condition = 0KSΔ . Since the experimental fact that the high-temperature asymptote is observed as (exp) 1 TC T −Δ ∝ , the AG model was speci- fied in both thermodynamic and dynamic aspects. Hence, the configurational entropy (8) was found [17] in the expli- cit interpolation form, namely (int) = 1 , with = ,K T T CS S S T B∞ ∞ ⎛ ⎞Δ Δ − Δ⎜ ⎟ ⎝ ⎠ (9) where B is the VFT-form dynamic parameter, defined in Eq. (2), and C is the thermodynamic parameter given in Eq. (10). Also, Eq. (7) was introduced into the AG model (see, e.g., [16]) via ( ) ( ) 10 ( ) ( )= , with = .log ln10 AG AG T T AG AG BT T nC CA kT S S Δμ τ + Δ Δ (10) 2.2. Fluctuation mechanism of cluster formation The idealized isobaric process of solid–cluster forma- tion is defined by the variation of the Gibbs potential (sol) (sol) (sol)( , ) = ( )T T TG n p S n T nδ − δ +μ δ . The cluster growth, if adopted at the thermal equilibrium temperature eqT , is determined by a minimization of the total system potential that requires (sol) (liq) eq eq=μ μ for the chemical potentials and (sol) (liq) eq eq<S S for the entropies, given at eq=T T [19]. The stabilization of the liquid–solid boundary, when temperature drops below the equilibrium temperature, is also ensured by the minimization of compe- Loosing thermodynamic stability in amorphous materials Fizika Nizkikh Temperatur, 2011, v. 37, No. 5 553 ting Gibbs potential and, requiring positivity of both the excess chemical potential (sol) (liq)=T T TΔμ μ −μ and entropy (liq) (sol)( ) = ( ) ( )T T TS n S n S nΔ − at eq<T T . Even though the SCL system is not globally at equilibrium, it can be di- vided into subsystems that are almost at equilibrium with their neighbors and TSΔ = = < ( ) >T CS nΔ can be intro- duced via the configurational average 0 < ... > = ... ( ) ,C TP n dn ∞ ∫ (11) through the local-equilibrium conditions. Within the fluctuation mechanism adopted for the for- mation of spatially heterogeneous clusters, the probability ( )TP n dn of meeting a solid cluster of molecular size n is that of finding the variable n lying between n and n dn+ . This process is driven by the temperature-dependent total system entropy (sol) (liq) (mix)( ) = ( ) ( ) ( )T T T TS n S n S n S n+ + , including the solid–liquid mixing term (mix) TS , passing through the maximum at = Tn n , in the quasi-static ap- proximation. The maximum-entropy principle [19] pro- vides the probability density 2 2 ( ) ( ) ( ) = exp exp , 2 T T T B T S n n n P n k n ⎡ ⎤⎡ ⎤ − −⎢ ⎥⎢ ⎥ Δ⎢ ⎥⎣ ⎦ ⎣ ⎦ ∼ (12) where the mean glassy-cluster molecular size Tn is estab- lished by the thermodynamic conditions of the cluster con- tact with the thermal bath of temperature eqT . The cluster- size fluctuation is introduced by 2 2= < ( ) >T T Cn n nΔ − = ( ) 12 2 = / T T B n n S k n − − ∂ ∂ , emerging in Eq. (12), that is expected to work beyond the equilibrium. The clusters of molecular size n appear and disappear from a fluid volume with frequency 1( )T n−τ , where ( )T nτ is lifetime of a given n-cluster. The probability of rear- rangement of such kind of clusters is given by 1( )T n−τ de- termined by the maximum of the boundary-formation en- tropy ( )TS nΔ established at the mesoscopic size-scale, namely 1 1 1 min min min eq ( ) ( ) ( ) = exp = exp .T T B B S n W n n k k T − − − ⎡ ⎤⎡ ⎤Δ τ τ τ −⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ (13) Here min = ( )TW G nΔ corresponds to the minimum work required for solidification of n molecules driven by clus- ter-size fluctuations [19]. Within the adopted isobaric- isothermal mechanism, one has (sol) (liq) eq ( ) =T TG n n ⎡ ⎤Δ μ −μ⎢ ⎥⎣ ⎦ . Consequently, Eq. (13) can be extended to supercooled eq( < )T T states through the relation ( )( ) = exp /'T a T Bn n k Tτ τ Δμ , introducing cluster relaxation time on the mesoscopic time- scale < ( ) >T T Cnτ ≡ τ , whereas < >T Cn n≡ . In the simplest case of Tn n≡ , when clusters-size fluc- tuations are ignored, one naturally arrives at pseudo- Arrhenius form ( )( ) ( ) min= exp /VFT VFT T BT E k Tτ τ , with =T T TE nΔμ , following from the VFT and AG forms presented in Eqs. (1) and (7). When the Gaussian fluctua- tions are large ( >T Tn nΔ ), the extension of the Arrhenius form via Eq. (13), namely (mod) = exp 1 , 2 T T aT B B E E k T k T ⎡ ⎤⎛ ⎞Δ τ τ +⎢ ⎥⎜ ⎟ ⎢ ⎥⎝ ⎠⎣ ⎦ (14) results in a high-temperature thermodynamic perturbation series for the primary relaxation scale [12]. When the mode coupling theory (MCT) [1] is extended by the energy fluctuations TEΔ [20], one obtains ( ) ( ) = exp 1 , for , 2 1 ET MCT k TBMCTe T a cT cB c E T T k T T T ∞ γ ⎡ ⎤⎛ ⎞Δ τ τ τ + ∝⎢ ⎥⎜ ⎟ ⎢ ⎥⎝ ⎠⎣ ⎦ ⎛ ⎞−⎜ ⎟ ⎝ ⎠ (15) from Eq. (14) providing asymptotically the known MCT extrapolation form [1]. When the thermal fluctuations of CRRs are taken into consideration, the extended version of the AG theory was introduced [20] as ( ) 0 0 1= exp 1g a g aAGe aT B T c B T n n k T k T ⎡ Δμ Δμ ⎤⎛ ⎞ τ τ +⎢ ⎥⎜ ⎟⎜ ⎟ε γ ε⎢ ⎥⎝ ⎠⎣ ⎦ , (16) where an is the lower limit for the CRR number Tn . One can see that both dynamic and thermodynamic forms are linked by the MCT slowing down exponent cγ , lying in the range [12] (exp)2 < 3c≤ γ for SCLs and (exp)3 < 4c≤ γ for glass-forming polymers. Also, the ratio for the excess chemical potentials was obtained (see Eq. (37) in [12]), namely = 1 8 ln10 1 4 g g g c c c m∗⎛ ⎞Δμ ε ⎜ ⎟+ −⎜ ⎟Δμ ε γ⎜ ⎟ ⎝ ⎠ , (17) where = 16 2gm∗ ± is the model-independent parameter. 3. Thermodynamic instability 3.1. Ergodic versus nonergodic version Exploring a minimum set of the observable parameters in glass-forming materials, i.e., 0T , gT , gm , cT , and cγ , Valery B. Kokshenev 554 Fizika Nizkikh Temperatur, 2011, v. 37, No. 5 the ergodic A-version for self-consistent description of the primary timescale (exp) Tτ , its steepness (exp) Tm , and curva- ture (exp) T [20] can be introduced. Specifically, the high- density solid-like ergodic clusters are presented in Eq. (16) by relation times ( ) ( )A AGe T Tτ ≡ τ . Even though the global equilibrium is not achievable in SCLs, the ergodic hypo- thesis has been in fact implicitly employed via the Gaus- sian cluster distribution ( ) ( )A TP n (12), providing the ther- modynamic description for long-living metastable states. In other words, it is suggested that thermodynamic ma- croscopic observables evaluated above eT in the thermo- dynamic limit N →∞ can be well approximated by cor- responding statistical quantities estimated in the static limit t →∞ . In order to describe (exp) Tτ below eT , the low-density solid-like clusters are introduced via a nonergodic B-ver- sion, with the help of non-Gaussian distribution ( ) 0 1( ) exp , < ,B eT T nP n T T T σ⎡ ⎤⎛ ⎞⎢ ⎥− ≤⎜ ⎟σ ζ⎢ ⎥⎝ ⎠⎣ ⎦ ∼ (18) presented by the Stauffer cluster scaling form [21] taken in the simplest large-cluster approximation. Here =Tζ 1/ b T − σ= ζ ε stands for the typical cluster molecular size, always exceeding bζ , and Tε is given in Eq. (1). The standard method of steepness descent results in the late- time primary relaxation given by 1( ) 1= expB b b a TT B Tk T σ σ− ⎡ ⎤ ⎛ ⎞Δμ ζ⎢ ⎥σ −⎛ ⎞τ τ ε ⎜ ⎟⎜ ⎟⎢ ⎥σ ε⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎣ ⎦ (19) obtained for large and stable clusters ensured, respectively, by the saddle-point conditions > 1σ and 2σ ≥ . Aiming to design a new nonergodic VFT–AG fitting form in terms of the percolation-type clusters specified by = 2σ , Eq. (19) is reduced to 2 ( ) ( ) 02 2= exp exp , for < , 2 B B b b e a a eT TB T n D T T T k T ⎛ ⎞ ⎛ ⎞Δμ τ τ ≈ τ ≤⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟εε⎝ ⎠ ⎝ ⎠ (20) where 2=b bn ζ and a new strength index ( ) =B eD 2 2/2( )b b B en k T= Δμ defined at eT . Thus, Eq. (18) emerges as a pseudo-Gaussian form with ( )2 = B T Tn nΔ 1 b Tn −≡ ε . The proposed distribution ( )B TP provides a description of the A–B-state crossover which is thought of as a smooth cluster structure rebuilding at eT . We therefore define con- tinuous changes in the cluster molecular size and variance via ( ) ( )=A B e en n and ( ) ( )=A B e en nΔ Δ . The variance closely related to the cluster structure exhibits a kink [20], when it transforms from ( ) ( ) 0= = / (1 / )A A c c aT Tn n n T TΔ ξ ξ − into ( ) 1/2 0= = / / 1B b T bTn n T T−Δ ε ζ − . Accounting for these relations, the conditions of cluster continuous changes yield 0 0 2 2= = , 1 b e a c a n T T T n n− ξ (21) where = 2 / .c cξ γ Also, the condition of continuity for the strength indexes, i.e., ( ) ( )=A B e eD D , where ( ) =A eD ( )AGe eD= is established with the help of Eq. (16), taken at = eT T , provides 2 0 2 0 0 11 = . 2 g a g a b c B e B e n n n T k T k T Δμ Δμ⎛ ⎞ Δμ +⎜ ⎟⎜ ⎟γ ε⎝ ⎠ (22) Here the metastable states A and B are energetically dis- tinguished through their chemical potentials TΔμ , approx- imated by constants within the corresponding domains: ( ) = =A a gTΔμ Δμ Δμ , for <e cT T T≤ , but ( ) =A cTΔμ Δμ , when cT T≥ ; ( ) 0= =B bTΔμ Δμ Δμ , for 0 < eT T T≤ . Then, employing Eq. (21), one obtains the relations 0=c a c cn TΔμ γ ε and 2 0= 2 / 2 /c c c a c e eT n TΔμ ε ξ = ε ε , re- ducing Eq. (22) to 2 2 0 0 0 0= ( ) , = , with < < .g e c e c c g T T T T T T Tμ μ Δμ − Δμ + − Λ Λ Δμ Δμ (23) To find ratios between the AGe chemical potentials in- volved in Eq. (23), one needs to parametrize the observed fragility data (exp) gm through the steepness functions ( )A Tm ( ( )= AGe Tm ) and ( )B Tm , obtained at gT and extrapolated to gT , via the corresponding model versions ( )A Tτ (16) and ( )B Tτ (20). As seen in Eq. (17), the AGe timescale parametrization involves, besides the chemical potentials, the slowing down exponent ( )A cγ . This allows one to reformulate the timescale fitting problem as follows. Instead of the obser- vation of fragility through the equations ( ) (mod) ( )=A B g g gm m m≈ with (mod) 1= (1 )g g gm m∗ −+ ε , we de- scribe the experimental data on (exp) cγ , within the frame- work of the A and B scenarios. They are introduced by the estimates 1 ( ) = ln10 1 2g gA c c c g c g c g m − ∗ ⎡ ⎤⎛ ⎞Δμ Δμε ε ⎢ ⎥γ +⎜ ⎟⎜ ⎟Δμ ε Δμ ε⎢ ⎥⎝ ⎠⎣ ⎦ (24) and 122 2 ( ) = ln10 1 2 ,gB e c e c g c g e g g mT m T m − ∗ ∗ ⎡ ⎤⎛ ⎞⎛ ⎞⎛ ⎞Δμ ε⎢ ⎥⎜ ⎟γ +⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟Δμ ε ε⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ (25) obtained through ( )A gm and ( )B gm , calculated by steepness Tm (3) at = gT T on the basis of corresponding ( )A Tτ (16) and ( )B Tτ (20). In Fig. 1, the numerical analysis of the model pre- dictions in Eqs. (24) and (25) is provided where unknown Loosing thermodynamic stability in amorphous materials Fizika Nizkikh Temperatur, 2011, v. 37, No. 5 555 temperature eT in ( )A cγ and ( )B cγ is excluded by means of Eq. (23). The observation in Fig. 1 of the experimental data (exp) cγ for the SCLs through the relations ( ) (exp) ( )=A B c c cγ γ ≈ γ results in the model estimates / = 1.32 0.05g cΔμ Δμ ± and 0 / = 1.47 0.06cΔμ Δμ ± . 3.2. Observation of the ergodic–nonergodic crossover The obtained in Fig. 1 estimates for the chemical poten- tials of solid-like clusters in SCLs provide ( ) = 0.317SCL μΛ in Eq. (23) for glass crossover temperature ( )SCL eT . Being numerically close to 1/ 3 , this result supports another es- timate ( ) 0= (2 ) / 3SCL e cT T T+ discussed in Eq. (42) in Ref. 12. Here, we specify Eq. (23) via Eqs. (5) with = 16gm∗ and Eqs. (6) with = 7cm∗ , via a new prediction 2( ) 13.5 76.5 = ( 7) SCL g ge g g g m mT T m m − + − (26) now extended from organic to inorganic liquids. In Fig. 2, Eq. (26) is analyzed in light of the data on the critical temperature cT , proposed earlier by Colby [9], which signals the dynamic instability below gT associated here with glass transition temperature eT . Taking also into consideration that no adjustable parameters are used in Fig. 2, we infer that the critical behavior observed in the dynamic experiments in SCLs [9,22] originates from the loss of ergodicity below gT , firstly reported in Ref. 23 and then explained in Ref. 6. Despite of the fact that the em- ployed AGe model is limited by “regular” liquids, Eq. (26) also includes salol, as follows from the analysis in Fig. 2. This implies that during the A–B crossover, differently distributed clusters expose a similar, model-independent behavior. In other words, the smallest glassy-like clusters, which emerge near eT , have a gas-like structure [6], re- gardless of the underlying chemical potentials and geome- try. Likely the same refers to the glassy-cluster-size fluctu- ations, which distinguish the “regular” and “irregular” liquids above eT discussed in Ref. 12. These observations suggest application of general Eq. (23) to other glass-forming materials, where μΛ is treated as a generic parameter. In Fig. 3, the special case of polymers is presented by 2(pol) 12.5 72.8 = . ( 7) g ge g g g m mT T m m − + − (27) For the case of spin-glass forming metallic alloys, a similar analysis was elaborated on the basis of kinetic data Fig. 1. Observation of the data on the slowing-down exponent in SCLs through the ergodic (a) and nonergodic (b) versions for the glassy-like solid clusters. a — The points are available experi- mental data on (exp) cγ taken from Table 1 in Ref. 12. The bars indicate experimental error. The dotted line is the best linear fit of the data. The solid line is drawn through Eq. (24), with = 16gm∗ and the fitting parameter / = 1.32g cΔμ Δμ . b — The dotted line is the same as in a. Points are model predictions for the slowing- down exponent (mod) cγ through Eq. (25). The solid line is drawn through Eqs. (23) and (25), with = 16gm∗ , / = 1.32,g cΔμ Δμ and the overall fitting parameter 0 / = 1.47cΔμ Δμ . PG m-TCP picoline PC glycerol salol TNB OTP a b γc γc 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 2-MTHF sorbitol toluene triphenphosfite PDE 3 BP dibutilphthalate n-propanol mg mg 30 40 50 60 70 80 90 100 40 60 80 100 120 140 160 Fig. 2. Observation of the ergodic–nonergodic crossover in su- percooled liquids. The solid line is the glass crossover tempera- ture eT reduced by gT , shown in Eq. (26). The dashed line is a linear interpolation of Eq. (26) to the strong-glass SCL regimes. The points, taken from Table 1 of Ref. 22, are the outcome of the fitting analysis [22] of the data on dielectric loss, viscosity, and mechanical shear relaxation in low molecular weight organics (open circles) and inorganic (closed circles) SCLs. l-propanol glycerol TNB OTP PC m-toluidine salol — Organic liquids — Inorganic liquids 0 20 40 60 80 100 120 0.95 0.90 0.85 0.80 0.75 B O2 3 GeO2 T T e g / mg Valery B. Kokshenev 556 Fizika Nizkikh Temperatur, 2011, v. 37, No. 5 on the relaxation timescale [9]. One finds (met) = 0.285μΛ , using the i) data on the critical temperature shifts c g cT T TΔ = − , with =c eT T , observed [9] as 75 and 72 K for two alloys 48 32 20Pd Ni P and 60 15 25Pt Ni P , having the common fragility = 48gm , and ii) the data [24] on the glass transformation temperature =gT 566 and 482 K, respectively. In turn, the generic parameter for metallic alloys results in the prediction for glass crossover tempera- ture 2 (met) 14.5 80.1 = ( 7) g g e g g g m m T T m m − + − (28) illustrated in Fig. 4. 4. Conclusion We have discussed the problem of the true thermody- namic instability occurring during structural glass and spin-glass transformation in glass forming materials pre- sented here by molecular, polymeric supercooled liquids and metallic alloys, respectively. It has been repeatedly demonstrated that a generalized solution to the glass for- mation process in microscopically different systems is possible, if designed on mesoscopic level. Although a unique coherent theoretical framework remains a chal- lenge, the proposed complex geometric, dynamic and thermodynamic approach to the problem offers novel rela- tions between the observable dynamic exponents and thermodynamic and kinetic macroscopic parameters. A macroscopic parametrization of the primary timescale, made in a self-consistent manner, provides a strong evi- dence for mutual dependence between thermodynamic 0(T and gT ) and dynamic ( cT ) characteristic temperatures [12,14] as well as ergodic–nonergodic temperature [6] eT . In this study, we have improved analysis of the critical temperature eT presenting it in the model-independent form that allowed one to extend the applications from the glass-forming organic and inorganic liquids over polymers and metallic glasses. All predicted crossover glass temper- atures indicate that the distance between the two characte- ristic temperatures gT and eT decreases when the glass material fragility grows, implying that true glass transition near the glass transition temperature gT can be expected only in the very strong glass formers. Finally, the proposed approach can be extended over traditional quadrupolar orientational glasses and modern dipolar orbital glasses, for which microscopic description is developed in Refs. 25 and 26, respectively. For these cases, an evaluation of the ergodic–nonergodic instability boundary is though limited by the absence of systematic data on the characteristic temperatures and slowing-down exponents discussed in Eqs. (23)–(25). Acknowledgments The financial support by CNPq and FAPEMIG is ac- knowledged. 1. W. Götze and L. Sjögen, Rep. Prog. Phys. 55, 241 (1992). 2. C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, and S.W. Martin, J. Appl. Phys. 88, 3113 (2000). 3. V.B. 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