Ergodicity failure near structural glass transformations

PERGAMON

Solid State Communications 119 (2001) 429±434

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Ergodicity failure near structural glass transformations V.B. Kokshenev* Departamento de Fisica, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, Minas Gerais, Brazil Received 11 November 2000; received in revised form 29 March 2001; accepted 6 June 2001 by C.E.T. Goncalves da Silva

Abstract Relaxation timescale for glass-forming materials is analyzed within a self-consistent description introduced within percolation theory treatment of the Adam±Gibbs model. The ergodic±non-ergodic phase diagram is proposed for the case of molecular supercooled liquids in terms of the ergodic-phase instability temperature TE vs fragility. TE, being below and close to the glass-transformation temperature Tg, is established through violation of the ergodic hypothesis, i.e. by a crossover from the Gaussian (`ergodic') to a non-Gaussian dynamics of evolution of clusters. The ®nite-size fractal cluster distribution is deduced from the known Stauffer scaling form. Crossover of the compact-structure (`ergodic') clusters to the hole-like glassy clusters is attributed to their critical-size thermal ¯uctuations. q 2001 Published by Elsevier Science Ltd. PACS: 64.70.Pf; 61.20.Lc Keywords: A. Disordered systems; D. Dielectric response; D. Phase transitions

The structural transformations observed under cooling cycles in glass-forming materials is essentially a crossover from a thermally equilibrated ergodic state to a metastable, non-ergodic state, characteristic of an amorphous solid. Researchers have long searched for a signature of the underlying `true' ergodic±non-ergodic transition emerging at a certain ergodic-instability temperature, say, TE. Examples are the known Adam±Gibbs (AG) model [1], in which TE ˆ T0 (Vogel±Fulcher (VF) temperature), and recently reported ideal glass models [2,3] …TE ˆ 0†; the known mode-coupling theory [4] (TE ˆ Tc ; crossover temperature that lies above the glass-transformation temperature Tg) and the ®rst- [5] …TE ˆ T0 † or the second[6] …TE ˆ Tc † order transition models. In this communication, we establish TE in amorphous materials through the ergodic hypothesis tested by observations within the scope of the percolation-theory treatment. The dramatic growth of the viscosity in supercooled liquids is related to seemingly divergent behavior of the slow relaxation timescale t T. The non-Arrhenius t T is wedged between two singularities, one algebraic and the other exponential. These were introduced by characteristic temperatures Tc and T0, respectively, material-dependent * Tel./fax: 155-31-3499-5600. E-mail address: valery@®sica.ufmg.br (V.B. Kokshenev).

slow-down exponent g c and material- and T-dependent strength parameter DT through the well-known [4] phenomenological ®tting forms, tT / …T 2 Tc †2gc and ln tT / T0 DT = …T 2 T0 †: As a result, the overall T-behavior of the timescale for a given glass former is documented by certain sets of ®tting parameters, such as characteristic temperatures [7,8] T0 < TK , Tg , Tc (TK is the Kauzmann temperature) [9] and of g c, Dg …ˆ DT at T ˆ Tg † and mg. The latter is Angell's fragility [7] (or steepness index [10]), 1 de®ned as mT ˆ 2d log10 tT =d ln T and derived from the observed t T at the glass-transformation temperature Tg. Exploring the sets of observable parameters, we speculate on ergodic (A) and non-ergodic (B) versions of the timescale description. These will be given in two explicit selfconsistent, asymptotic-type forms introduced away from singularity points T0 and Tc, and parametrized between them at point Tg through the observed mg. We employ the percolation-theory treatment of the second-order transition [1,6,11±13] by which the underlying ideal liquid±glass transition occurs when an in®nite solid-like cluster emerges 1

One can see that the effectiveness of the strong-fragile glass classi®cation is due to weak T-dependence of DT near Tg, but its strong material dependence found in explicit form, Dg ˆ …mp1 †2 ln 10=…mg 2 mp1 † (see Eq. (4) in Ref. [7]), given by the material-dependent fragility mg and the `universal' parameter mp1 :

0038-1098/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII: S 0038-109 8(01)00270-8

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V.B. Kokshenev / Solid State Communications 119 (2001) 429±434

at the VF temperature T0. It has been demonstrated [14] that percolation on large, but ®nite, statistically independent fractal clusters, works successfully to describe the primary relaxation observed in distinct classes of amorphous materials, such as supercooled liquids, amorphous polymers, canonical and orientational glasses [15]. Generic features of the order±disorder phase transitions were ®rst introduced rigorously [16] into the percolation theory through the distribution function PT(z) (for a ®nite cluster of size z) on the basis of a percolative free-energy analog of the Ising model. As a result, it was recognized that PT exhibits a singularity near the percolation threshold, and shows a crossover from the exponential form, characteristic of the disordered high-T phase to the stretched exponential form in the (long-range) ordered phase. Moreover, the crossover characteristics of PT were much used in the liquid± glass transition problem [17±19]. Very recently, the slowrelaxation (near Tg) and fast-relaxation (near Tc) mechanisms were derived [20] from the dynamical spectra of glass-forming polymers and liquids, respectively, on the basis of pseudo-Gaussian forms established earlier for PT(z) within a number of restricted-diffusion theoretical models. In this study, we demonstrate how the characteristic features of the low temperature …T , TE † metastable `B' phase, are integrated into percolation theory through PT…B† for fractal df…B† -dimension clusters (df…B† , d; where d is the percolation lattice dimension [21]). In the ergodic phase A, the structural relaxation is associated with a thermally activated process, directed toward the equilibrium by the entropy. According to the fundamentals of statistical physics (Onsager's principle) [22] PT…A† must be Gaussian, at least near the equilibrium. 2 In line with the ergodic hypothesis, instability of A phase, introduced at T . TE . T0 with the help of non-fractal …df…A† ˆ d† clusters distributed by P…A† T ; is manifested by a crossover at T ˆ TE from the Gaussian to some non-Gaussian, say, PT…B† form. Such crossover dynamics near Tg was discussed in Refs. [18,12]. Experimental evidence for the Gaussian-to-Poisson PT crossover was given by Chamberlin et al. in Ref. [18]. We establish the P…B† form with the help of the percolation T theory ®ndings on the basis of a mesoscopic treatment of the AG model. The model of Adam and Gibbs suggests [1] that the VF temperature T0 …ˆ TK † of the underlying static phase transition is driven by the T0-vanishing con®gurational entropy DST of cooperatively rearranging regions, hereafter clusters. Thus, DST is introduced by the liquid-over-solid excess heat R capacity DCp(T ), i.e. DST ˆ TT0 …DCp …T†=T† dT: In accordance with the observed calorimetrical data [10], DCp …T† , T 21 ; we give a macroscopic VFT-type (Vogel± Fulcher±Tamman) representation of the AG model in the

2 Notably, it is justi®ed by analyses of the dynamical response given [18,19] for supercooled liquids.

well-known exponential form, namely,   mz za tT , ta exp a T with zT ˆ T T 12 0 T

…1†

with D0 ˆ ma za =T0 : Here we consider that zT , NA =DST (see e.g. Eq. (3) in Ref. [10]), where NA is the Avogadro's number; m a is a model excess chemical potential related to a typical cluster composed of zT molecules; t a and za represent the timescale and sizescale unit parameters. As demonstrated by Richert and Angell (by the combined analysis of the data on excess-heat capacity and dynamical response for a number of supercooled liquids) the compatibility of Eq. (1) with the observed t T is limited by the range Tg & T , TB < Tc (see Table 2 in Ref. [10]). Mesoscopic description of slow relaxation is based on the mesoscopic presentation of the AG model given by an ensemble of thermally relaxing, self-similar clusters of size z, i.e. tT …z† ˆ ta exp …mT z=T† and the cluster distributions PT…A† …z† , exp …2…z 2 zT †2 =2Dz2T † and PT…B† …z†: Here we extend the energetic description of a typical cluster by T-dependent chemical potential m T, and introduce its size 1=2 variance, i.e. DzT ˆ k…z 2 zT †2 lA ; given by PT…A† …z† through elaboration of a con®gurational average denoted by angle brackets k¼lA. Straightforward evaluation of tT…A† ˆ ktT …z†lA provides the macroscopic timescale in the ergodic phase, namely,    E DET for T $ TE tT…A† , ta exp T 1 1 …2† T 2T with ET ˆ mT zT and DET ˆ mT Dz2T =zT : Eq. (2) not only reproduces the VFT form (Eq. (1)) in the small size-¯uctuation limit, DzT p zT ; but represents an extension of the AG-model timescale above Tc. Expanding the exponential (Eq. (2)) into the high-T series, we represent it approximately as an algebraic, effectively divergent asymptotic form, namely,

tT…A†



DET 1¼ / ta 1 1 2T

Tc ˆ T0 1

gc ˆ

mc za gc

 ET T

< 

ta  ; T q Tc T gc 12 c T

and

m c za 2 T ˆ 2 with 1T ˆ 2 1; T0 1c T0 ja

…3†

where Tc ˆ DEc =2 and gc ˆ Ec =Tc are found by a comparison of the leading terms of the exponential and the algebraic series. Also, the model relation, DzT ˆ j a zT ; is adopted for the A-phase. Thus, we arrive at the algebraic form tT / …T 2 Tc †2gc ; induced by the second-order phasetransition theory and used much as a phenomenological ®tting form far away from Tc, where no true divergency is observed (see discussion and analyses in Figs. 13±15 in Ref. [4]).

V.B. Kokshenev / Solid State Communications 119 (2001) 429±434

The analyses given in Ref. [10] clearly show that, besides the temperature region T . Tc ; the VFT representation (Eq. (1)) of the AG model is not valid below Tg and, therefore, a non-ergodic VFT version of the AG model, that works at T0 , T # Tg ; is needed. When T approaches T0, Stauffer's scaling form (see Eq. (2.22) in Ref. [21]), PT…B† …z† / exp 2 …1=s†…z=z T †s ; can be employed 3 for large fractal clusters, z q zT : Here zT ˆ zb 1T2…1=s† ; with 1 T given in Eq. (3), stands for a size of a typical cluster, that percolates through the lattice of constant z b. Straightforward application of the standard saddle-point method yields

tT…B† , ta exp

  s s21 m z s21† 1T b b s T 1T

…4a†

for the large …s . 1† and stable …s $ 2† fractal clusters of mean size z T…B† ˆ zb …mb zb =T 1T †1=…s21† : Compatibility with the phenomenological VFT form requires s ˆ 2: Thus, for the `non-ergodic' part of the timescale, we have

t T…B† , ta exp

m2b zb for TE $ T . T0 : 2T 2 1T

…4b†

We see that P…B† T …z† is given by a pseudo-Gaussian form 2 with Dz2T ˆ zT…B† ˆ zb 121 T and zb ˆ zb : Therefore, the A- and B-phase generalized distribution PT(z) preserves the Gaussian form near the crossover temperature TE but the variance of random-size clusters exhibits a kink, i.e. transforms DzT from DzT…A† ˆ j a za =…1 2 T0 =T† to DzT…B† q ˆ zb…B† =…T=T0 2 1†: The continuity of the distribution PT(z) at a crossover point T ˆ TE ; given by a couple of the mesoscopic-scale equations, zE…A† ˆ z…B† and Dz…A† E E ˆ …B† DzE ; yields TE ˆ T0

zb T0 ˆ : za 1 2 za j a2

…5†

One more macroscopic equation results immediately from the continuity of the VFT-like forms given by DE…A† ˆ D…B† E ; namely,   m g za 1 m g za m2 z 11 ˆ 0 2b : T0 gc T0 1E 2TE

…6†

On the other hand, A and B phases are presumably distinguished by their chemical potential m T. Within the domains T0 # T # TE ; TE # T , Tc and T $ Tc we introduce …A† different AG-model parameters m…B† T ˆ mb ˆ m0 ; mT ˆ …HT† ma ˆ mg and mT ˆ mc ; respectively. From Eqs. (3) and (5) one ®nds relations mc za ˆ gc 1c T0 and mc ˆ 3

By percolation models [21] s ˆ 1=nd f and n is the correlationlength critical exponent related to the fractal cluster radius-size Rp ˆ R0 up 2 p0 u2n or the volume-size number zp , Rdpf near the threshold p0. The T-dependence is introduced by the evident relations p…T† 2 p0 < …T 2 T0 †p 00 , 21T :

21c T0 =za j a2

431

ˆ 21c TE =1E which reduce Eq. (6) to

TE ˆ T0 1 …Tc 2 T0 †

m20 2 m2g for T0 , TE , Tc : mc mg

…7†

We complete the parametrization of the timescale by using the fragility data mg. Straightforward estimations of mT with the help of Eqs. (2) and (4) provide two different …B† equations m…A† T and mT that involve all the parameters introduced above. These equations to be considered at T ˆ Tg and formally solved for the slow-down exponent (Eq. (3)) are

g…A† c ˆ

mp1 ln 10 ! if Tg $ TE; mg 1c mg 1c 112 mc 1g mc 1g

…8†

and

g…B† c ˆ 

mp1 ln 10 !2 ! if Tg # TE : …9†  m0 2 12c TE mp1 112 mc 1g 1E Tg mg

Here we adopted for the observed fragility the relation p 1 [7], mg ˆ mp1 …1 1 121 g †; where m1 is a constant. We see that the slow relaxation data for the timescale, introduced by mg, are parametrized by the two sets of parameters {T0, Tg, Tc} and {m 0, m g, m c}. Eqs. (8) and (9) are considered as two distinct predictions for the observed slow-down exponent g c according to the ergodic and nonergodic scenarios of a timescale T-evolution, respectively. We want to stress that our aim is not to improve the existing description of the observable relaxation timescale or to establish all AG-model parameters. Conversely, by means of the self-consistent mesoscopic description of t T we are going to establish an ergodic±nonergodic boundary. Differently, with the help of Eqs. (8) and (9) we put a question: whether the observed Tg belongs to the A-phase …Tg . TE † or to the B-phase …Tg , TE †? A comparison with experiment is elaborated on the basis of the following axillar relations Tg mg ˆ T0 mg 2 mp1

and

mg 1 mp2 Tc ˆ Tg mg 2 mp2

…10†

with mp1 ˆ 16 ^ 2 and mp2 ˆ 7 ^ 1; where deviations are due to scattering of experimental data (see Table 1 in Ref. [8]). Existence of Eq. (10) implies that the characteristic temperatures are not independent [8], which permits us to eliminate them from Eqs. (7)±(9) and to directly employ Angell's mg-classi®cation for glass formers. In Fig. 1, the ratio mg =mc is established by ®tting gc…A† (Eq. (8)) with experimental data g c (shown by A lines) for a number of molecular liquids and polymers. Then, ®tting gc…B† (Eq. (9)) with the help of Eq. (7) and the established ratio mg =mc (shown by B lines in Fig. 1) provides the overall ratio m0 =mc for distinct families of glass formers. We see that the found AG-model chemical potential m T decreases with

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V.B. Kokshenev / Solid State Communications 119 (2001) 429±434

Fig. 1. Slow-down exponent of the relaxation timescale against fragility. Points Ð experimental data (see Table 2 in Ref. [20]) for molecular liquids (PC Ð propylene carbonate, m-TCP Ð m-tricresyl phosphate, OTP Ð ortho-terphenyl, PG Ð propylene glycol) and amorphous polymers (PS Ð polystyrene, PET Ð poly(ethylene terephtalate), PMA Ð poly(methyl acrylate), PBD Ð polybutadien deuterated, n-BB Ð butylbenzene, PPG Ð poly(propylene glycol)). Lines Ð the best ®tting of the ergodic (A, Eq. (8)) and nonergodic (B, Eqs. (7) and (9)) versions by the adjustable Adam±Gibbs model parameters mg =mc ˆ 1:35 (and 1.08) m0 =mc ˆ 1:50 (and 1.70) for liquids (and polymers).

temperature, i.e. m0 . mg . mc for T0 , Tg , Tc ; that is consistent with the growing of the observed con®gurational entropy [10]. In the case of polymers, one can speculate from Fig. 1 that TE , Tg : On the other hand, it is known [19] that the chain-like structured polymers are fractal clusters …df , d† even in the ergodic phase and, thus, the proposed scheme becomes inconsistent. In the case of the supercooled liquids an inconclusive relation TE < Tg follows from the analysis given in Fig. 1. Nevertheless, the set of reduced parameters m 0, m g and m c, established within the approximation TE < Tg permits us to improve the estimation of TE through Eq. (7) and to propose the ergodic±nonergodic phase diagram shown in Fig. 2. We see that the dynamical instability of the ergodic phase occurs close and below the glass-transformation temperature Tg. This conclusion results from the given description of the observed fragility [7] (with accounting of the experimental accuracy shown by dash-dotted lines and, approximately, by the size of points in Fig. 2) and is in a qualitative agreement

with the ®ndings reported in Ref. [18] on the signature of the ergodic±nonergodic transition at TE o Tg derived from the stress-relaxation data (in ionic liquids) and from the dielectric loss spectra (in salol). In summary, we have discussed structural slow relaxation in amorphous materials through the observed timescale t T. Macroscopic consideration is given in terms of the characteristic energetic and geometrical parameters common of the AG model, namely, the one-molecule excess thermodynamic potential m T and the number-molecule cluster size zT. Mesoscopic description of the problem is reduced in practice in terms of the thermally relaxing clusters of characteristic time t T(z). The macroscopic timescale is introduced by a con®gurational average tT…C† ˆ ktT …z†lC (C ˆ A, B) carried out by relevant cluster-size distribution. We extended the VFT version of the AG model, limited [10] by T . Tg ; to the nonergodic B-phase for T , TE on the basis of percolative cluster treatment. The ergodic± nonergodic crossover temperature TE is de®ned through

V.B. Kokshenev / Solid State Communications 119 (2001) 429±434

433

Fig. 2. Ergodicity against fragility. Adam±Gibbs-model prediction for the ergodic instability line in glass-forming liquids (circles) and molten salts (diamonds) given for ZnCl2 Ð zinc chloride, CKN Ð calcium potassium nitrate, POC Ð phenyl-o-cresol (for others see Fig. 1). Solid lines are the ergodic-phase boundary TE, given by Eq. (7) with the adjustable parameters derived from Fig. 1; also, the reduced characteristic temperatures T0 and Tc are shown. Dash-dotted lines show the error scale (approximately, ^5%) caused by the scattering of the differentmethod experimental data. This is almost twice as large as the experimental accuracy when characteristic temperatures are derived within a given method (see also Fig. 1 in Ref. [8]).

violation of the ergodic hypothesis, namely, by a crossover of t…A† T (Eq. (2)), equivalent to the thermal average, elaborated by the Gaussian P…A† T …z† distribution for solid-like clusters, to the timescale t…B† (Eq. (4)), given by pseudoT Gaussian PT…B† …z† distribution for glassy-like clusters. The Poisson P…B† T …z† form is established through the Stauffer scaling theorem with an effective parameter ndf ˆ 1=2; consistent with the VFT form. As a result, the ergodic±nonergodic phase diagram is proposed in terms of observable variables in Fig. 2 and given for glass-forming simple (PC, m-TCP), complex (OTP, salol, POC, toluene) molecular and alcoholic (glycerol, PG, n-propanol) liquids. Molten salts (ZnCl2 and CKN) are also included because of their cluster-growth kinetic mechanism seems to belong to the class of molecular liquids (see Fig. 2 in Ref. [14]). We have demonstrated that the cluster-size ¯uctuations, unlike the cluster mean size, signal a crossover from the

ergodic phase to the nonergodic phase near TE. To elucidate this deduction, let us consider the cluster-volume-averaged space-density correlation function, lT ˆ Dz2T =zT 2 1 (see e.g. Eq. (118.6) in Ref. [22]). Thereby, the nonergodic …T0 , T , TE † and ergodic …Tc . T . TE † supercooled liquid states, as well as the `ideal ergodic' high-T liquid …T q Tc † are distinguished by lT…B† ˆ 0; lT…A† ˆ j a2 zT…A† 2 1 , 0 and l…HT† ˆ 21 …z…HT† ; 0†; respectively. We see that a T T structure of `nonergodic' clusters, given by Poisson's distrip bution PT…B† …z† with a size ¯uctuation DzT ˆ zT ; is characteristic of the ideal-gas [22,23] and, therefore, should be associated with a weak-correlated molecule structure. Again, accounting the thermodynamical meaning of l T, given through the cluster compressibility x T, one has lT < xT =xB 2 1; where x B corresponds to x T of the ideal-gas. We see that the growing of the solid-like clusters in liquid state under cooling is limited by their ¯uctuations. Instability of

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V.B. Kokshenev / Solid State Communications 119 (2001) 429±434

the supercooled liquid ergodic phase occurs when the cluster-size ¯uctuations reach maximum value common of the ideal gas. Therefore, ergodic-to-nonergodic structural transformation in supercooled liquids is characterized by reconstruction of solid-like clusters from a compact structure to the fractal …df , d† holey-like structure. Acknowledgements The author is grateful to C.A. Angell and R.V. Chamberlain for numerous fruitful interactions and for drawing the author's interest to the Adam±Gibbs model study. Thanks are due to Prof. R. Dickman and to Prof. P. Licinio for helpful comments on the manuscript. This work is supported by the Brazilian agency CNPq. References [1] G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. [2] D. Kivelson, G. Tarjus, X. Zhao, S.A. Kivelson, Phys. Rev. E 53 (1996) 751. [3] M. Schulz, Phys. Rev. B 57 (11) (1998) 319. [4] W. GoÈtze, L. SjoÈgren, Rep. Prog. Phys. 55 (1992) 241. [5] T.R. Kirkpatrick, D. Thrumalai, P.G. Wolynes, Phys. Rev. A 40 (1989) 1045.

[6] J. Souletie, Physica A 201 (1993) 30. [7] R. BoÈhmer, K.L. Ngai, C.A. Angell, D.J. Plazek, J. Chem. Phys. 99 (1993) 4201. [8] V.B. Kokshenev, Physica A 262 (1999) 88. [9] C.A. Angell, Physica D 107 (1997) 122. [10] R. Richert, C.A. Angell, J. Chem. Phys. 108 (1998) 9016. [11] J.P. Sethna, J.D. Shore, M. Huang, Phys. Rev. B 44 (1991) 4943. [12] T. Odagaki, Phys. Rev. Lett. 75 (1995) 3701. [13] V.N. Novikov, E. RoÈssler, V.K. Malinovsky, N.V. Surotsev, Europhys. Lett. 35 (1996) 289. [14] V.B. Kokshenev, Phys. Rev. E 57 (1998) 1187. [15] V.B. Kokshenev, N.S. Sullivan, J. Low Temp. Phys. 122 (3/4) (2001). [16] H. Kunz, B. Souillard, Phys. Rev. Lett. 40 (1978) 133. [17] M.H. Cohen, G.S. Grest, Phys. Rev. B 24 (1981) 4091. [18] R.V. Chamberlin, R. BoÈhmer, E. Sanchez, C.A. Angell, Phys. Rev. B 46 (1992) 5787. [19] R.V. Chamberlin, Europhys. Lett. 33 (1996) 545. [20] V.B. Kokshenev, N.S. Sullivan, Phys. Lett. A 280 (2001) 97. [21] M.B. Isichenko, Rev. Mod. Phys. 64 (1992) 961. [22] L.D. Landau, E.M. Lifshitz, Statistical Physics, Pergamon Press, London, 1989. [23] Microscopic realisation of the VFT-type relaxation, given in Eq. (4) with s ˆ 3, was proposed by J.T. Bender and M.F. Shlesinger through the defect-aggregation mechanism in J. Stat. Phys. 53 (1988) 37.