Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature

The structural relaxation in glass forming materials is studied near the glass transformation temperature Tg indicated by the heat capacity maximum. The late-time asymptote of the Kohlrausch–Williams–Watts form of the relaxation function is rationalized via the mesoscopic-scale correlated regions...

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Published in:	Физика низких температур
Date:	2007
Main Author:	Kokshenev, V.B.
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Language:	English
Published:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2007
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Cite this:	Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature/ V.B. Kokshenev // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 805-813. — Бібліогр.: 34 назв. — англ.

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spelling	Kokshenev, V.B. 2017-06-16T08:15:58Z 2017-06-16T08:15:58Z 2007 Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature/ V.B. Kokshenev // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 805-813. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 61.41.+e; 61.43.Fs; 64.70.Rf https://nasplib.isofts.kiev.ua/handle/123456789/121798 The structural relaxation in glass forming materials is studied near the glass transformation temperature Tg indicated by the heat capacity maximum. The late-time asymptote of the Kohlrausch–Williams–Watts form of the relaxation function is rationalized via the mesoscopic-scale correlated regions in terms of the Debye-type clusters following the dynamic scaling law. It is repeatedly shown that regardless of underlying microscopic realizations in glass formers with site disorder the structural relaxation is driven by local random fields, described via the directed random walks model. The relaxation space dimension ds = 3 at Tg is suggested for relaxing units of fractal dimension d f = 5/2 for quadrupolar-glass clusters in ortho–para hydrogen mixtures, that is compared with entangled-chain clusters in polymers (d f = 1) and solid-like clusters relaxing in supercooled molecular liquids (with ds = 6 and d f = 3). The relaxation dynamics of orientational-glass clusters in plastic crystals is attributed to the model of continuos time random walks in space ds = 6. As a by-product, the expansivity in polymers, molecular liquids and networks is predicted. The financial support by CNPq is acknowledged. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Related Topics Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature
spellingShingle	Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature Kokshenev, V.B. Related Topics
title_short	Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature
title_full	Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature
title_fullStr	Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature
title_full_unstemmed	Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature
title_sort	cluster relaxation dynamics in liquids and solids near the glass-transformation temperature
author	Kokshenev, V.B.
author_facet	Kokshenev, V.B.
topic	Related Topics
topic_facet	Related Topics
publishDate	2007
language	English
container_title	Физика низких температур
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
format	Article
description	The structural relaxation in glass forming materials is studied near the glass transformation temperature Tg indicated by the heat capacity maximum. The late-time asymptote of the Kohlrausch–Williams–Watts form of the relaxation function is rationalized via the mesoscopic-scale correlated regions in terms of the Debye-type clusters following the dynamic scaling law. It is repeatedly shown that regardless of underlying microscopic realizations in glass formers with site disorder the structural relaxation is driven by local random fields, described via the directed random walks model. The relaxation space dimension ds = 3 at Tg is suggested for relaxing units of fractal dimension d f = 5/2 for quadrupolar-glass clusters in ortho–para hydrogen mixtures, that is compared with entangled-chain clusters in polymers (d f = 1) and solid-like clusters relaxing in supercooled molecular liquids (with ds = 6 and d f = 3). The relaxation dynamics of orientational-glass clusters in plastic crystals is attributed to the model of continuos time random walks in space ds = 6. As a by-product, the expansivity in polymers, molecular liquids and networks is predicted.
issn	0132-6414
url	https://nasplib.isofts.kiev.ua/handle/123456789/121798
citation_txt	Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature/ V.B. Kokshenev // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 805-813. — Бібліогр.: 34 назв. — англ.
work_keys_str_mv	AT kokshenevvb clusterrelaxationdynamicsinliquidsandsolidsneartheglasstransformationtemperature
first_indexed	2025-11-25T20:40:25Z
last_indexed	2025-11-25T20:40:25Z
_version_	1850526284645400576
fulltext	Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7, p. 805–813 Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature V.B. Kokshenev Departamento de Física, Universidade Federal de Minas Gerais, Instituto de Ci �encias Exatas Caixa Postal 702, Belo Horizonte 30123-970, Brazil E-mail: valery@fisica.ufmg.br; valery.kokshenev@gmail.com Received October 5, 2006 The structural relaxation in glass forming materials is studied near the glass transformation temperature Tg indicated by the heat capacity maximum. The late-time asymptote of the Kohlrausch–Williams–Watts form of the relaxation function is rationalized via the mesoscopic-scale correlated regions in terms of the Debye-type clusters following the dynamic scaling law. It is repeatedly shown that regardless of underlying microscopic realizations in glass formers with site disorder the structural relaxation is driven by local ran- dom fields, described via the directed random walks model. The relaxation space dimension ds � 3 at Tg is suggested for relaxing units of fractal dimension d /f � 5 2 for quadrupolar-glass clusters in ortho–para hy- drogen mixtures, that is compared with entangled-chain clusters in polymers (d f �1) and solid-like clusters relaxing in supercooled molecular liquids (with ds � 6 and d f � 3). The relaxation dynamics of orientational-glass clusters in plastic crystals is attributed to the model of continuos time random walks in space ds � 6. As a by-product, the expansivity in polymers, molecular liquids and networks is predicted. PACS: 61.41.+e Polymers, elastomers, and plastics; 61.43.Fs Glasses; 64.70.Rf Glass transitions. Keywords: cluster, relaxation, fitting forms, fractal cluster, fractal cluster treatment, thermodynamic instability. I. Introduction The process of structural-glass transformation in supercooled liquids (SCLs) is followed by the formation of intermediate metastable states in which a dramatic in- crease in viscosity. These states expose anomalous tem- perature behavior of transport characteristics commonly studied above the glass transformation temperature Tg , established by scanning calorimetry [1]. An intriguing as- pect in glass transformation is the apparent connection between dynamics and thermodynamic features [2]. In particular, there is a great interest in complex, experimen- tal and theoretical, studies of the temperature-temporal behavior of primary structural relaxation in SCLs [1], which is often similar to that in other glass formers [3–8]. Dynamic data on the relaxation timescale � T (exp ) deter- mined in viscoelastic, dielectric, conductivity, mechani- cal relaxation, light and neutron scattering experiments is regarded as one of the main keys to the understanding of mechanisms of the structural glass transformation [1,9,10]. A unified approach to the problem given within one coherent framework remains a challenge for theorists [11]. Here we put forward a mesoscopic-scale conside- ration of the primary relaxation observed during vitri- fication in liquids and solids and approached by per- colative-geometric [12], dynamic-stochastic [8] and thermodynamic-statistic [13] treatments. In this study we focus on the mechanisms of the primary in solid and liq- uid glass forming materials specified by space-relaxation dimensions fractal clusters. We also stress the relation- ships between thermodynamic and dynamic observable quantities, which are raised from the underlying con- straints imposed on degrees of freedom in glass formation systems. Similarly to the case of the universal equation established for characteristic temperatures [13], the uni- versal mechanisms of relaxation will be found for distinct glass formers, presented here by SCLs, polymers, orien- tational quadrupolar-glass molecular, and spin-glass me- tallic and nonmetallic solids. The paper is organized as follows. In Sec. 2, the experimental and theoretical pre- liminaries are provided for dynamical macroscopic char- acteristics commonly describing non-Arrhenius tempera- © V.B. Kokshenev, 2007 ture and non-Debye temporal behavior in glass forming materials. Predictions for thermal expansion which are given in terms of the dynamical and thermodynamical (characteristic temperatures) parameters are provided in Sec. 3. The mechanisms of the primary relaxation near the glass transformation temperature are discussed in Sec. 4 for distinct glass formers, including a special case of the ortho-para-H2 mixtures. Conclusions are summarized in Sec. 5. 2. Preliminary 2.1. Fitting forms The phenomenological Vogel–Fulcher–Tammann (VFT) fitting form, namely � � T VFT gD T T T ( ) min exp� � � � � � � � 0 0 (1) is widely used to describe non-Arrhenius temperature be- havior of the structural relaxation in amorphous liquids and solids; Dg is the so-called strength index [14] and T0 is the VFT temperature. Proposed in the 1920s [9], Eq. (1) performs well within the temperature range established as T T Tg c � [13,15]. Here Tc is the crossover temperature Tc , which separates the moderately and strongly SCL states [13], distinguished in mode coupling theory (MCT) [1]. The pre-factor � min (exp ) � � �10 14 2 s reflects the Debye molecule vibrational times, characteristic of the nor- mal-liquid state. Besides Eq. (1), the non-Debye time-de- cay of structural correlations is also a generic feature of collective relaxation dynamics in supercooled states of all glass formers. The late-time dynamical response function is commonly fitted through the phenomenological Kohlrausch–Williams–Watts (KWW) form by the two temperature-dependent parameters T and �T [1]. Near Tg , the slow part of the relaxation function is due to the late-time asymptote � � S T t T t g (exp ) ( , ) exp ,� � � � �� for t g�� (2) Here g is the KWW stretching exponent that commonly weakly depends on T in the vicinity of Tg . The process of glass formation is followed by the strengthening of dynamic correlations. As the tempera- ture approaches Tg from above, it evolves smoothly start- ing from the Debye behavior typical of the normal liquid state [16,18,19]. The relaxation function can be therefore approximated by the ensemble of modified Debye-type clusters. The trial ensemble of clusters of random ra- dius-size R and relaxation time �D R( ) is characterized by the dynamically stable Debye clusters with the mean size RT and relaxation time �DT . Thus, �T t( ) gradually changes from the high-T Debye form to that given as � � ( ) ( ) ( , ) exp ( ) ( ) ,mod mod t T t R P R/R dR D T T� � � � � � � � � � 0 (3) where P R/R P R T T T ( ) ( ) ( , ) mod � stands for the radius-size distribution function for relaxing structural units. The lat- ter are solid-like clusters in the case of SCLs. Following the idea of the phase-ordering kinetics [20], the dynamic scaling law � �D T DT T z R/R R R g ( ) � � � �� , with R RT a� (4) was employed [12] near Tg , with the help of the dynamic cluster-dimension exponent zT for clusters restricted by the minimum cluster size Ra . 2.2. Fractal cluster treatment A schematic scenario of the stretched-time relax- ation can be figured out as a percolation process of fractal clusters of fixed structure, performing a diffusive motion at a given temperature T . The cluster-volume distribution density function P V T( , ) is thought of to be distinct for small-volume (V VT� ) clusters and for large-volume ( )V VT� clusters distinguished by the typical-cluster volume VT . The radius-size cluster statistics of self-similar clusters is therefore described by P R/R P V T dV/dRT T( ) ( , )� . The clus- ter fractal dimension d f is defined through [21] V V R R T T d f � � � �� . (5) In this way, the process of structural relaxation is treated as a percolation of correlated sites or bonds [21]. The site-percolation relaxation can be described through the model density probability [12] P x x x x R R u d dT ud d T f s ( ) ~ ~ ( ) exp ( ), , mod � � � � �1 . (6) It is introduced through the two cluster-shape parameters: d f (5 ) and d s , which is the effective dimension of space of late-time relaxation attributed to large clusters compet- ing with small clusters of size R RT� . In order to establish the model parameter u for distinct glass-formers, we have performed the model-independent analysis of dielectric response data in SCLs proposed by in Ref. 16. The well known Dixon–Nagel universal mas- ter curve [16] scales the dielectric susceptibility spectrum over a wide temperature range above Tg and over more than ten orders of magnitude in frequency�. In order to fit the Dixon–Nagel curve, and thereby to describe the ex- perimental data on dielectric susceptibility, we estimated the model susceptibility 806 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 V.B. Kokshenev � � � ��T T T D P x i x dx ( ) ( ) ( ) ( ) ( ) ( ) mod mod � � � �0 1 0 (7) which is a Fourier analogue of Eq. (3 ); �T ( )0 is the static susceptibility. The extended analysis of the Dixon–Nagel master curve provided on the basis of the model distribu- tion function P T ( )mod (6 ) and the Debye-cluster relaxation time �D R( ) (4) is similar to that described in Fig. 1 in Ref. 12. The results [17] u ug g.( ) ( ). . .,pol netw� � � �0 33 0 03 0 26 0 04, and u g liq � �0 25 0 06. . (8) obtained, respectively, for supercooled polymeric, net- work, and molecular liquids will be employed below for the cluster-dimension description in these materials. Another simple-model estimation for the slow-relax- ation function � � � � � S t T x d x x (mod) ( , ) exp [ ( )] exp [ ( )] ( ) � � � � � � !! 0 0 0 , (9) follows from Eq. (3). It is presented with the help of the auxiliary function� , the variable x R/RT� , and the tem- poral parameter �� t/ DT . Application of the standard method of steepness descent leads to the late-time asymp- tote, shown in Eq. (9) for �� 1, where !!� stands for the second-order derivative with respect to x. The saddle point x z/d f / z d f 0 1 � " ( ) ( ) is established by the station- ary condition [ ! �� ( )x 0 0, !! �� ( )x 0 0 ] which is valid for late times t d \|/zDT f /�� (\| )1 1 . Then, through a com- parison of Eq. (9) with the experimental data (2), one finds the model estimate for the stretching exponent re- sulting in the model relation [12] g s g s d z d (mod) � " . (10) When the effective dimension d s and the stretching expo- nent g can be established independently, Eq. (10 ) pro- vides the cluster-growth exponent prediction z dg s g (mod) � � � � � � � � 1 1 . (11) Also, a relation between the primary relaxation time and the mean intrinsic time of solid-like Debye clusters: � � T DT g g / g� � � ( )1 1 1 (12) follows from the simplest model (9), where �T and g can be observed by means of the KWW fitting form. Using � �DT a T a z R /R g� ( ) in the scaling form ( 4) and taking into account Eq. (12), the timescale steepness at Tg (fragility) estimates as m d d T m zg T T T z g g � � � � � � � � � lg ln * � with m d R d T z T T Tg * lg ln � � � � � � � � . (13) This provides a link between the fragility mg and the clus- ter-dimension dynamic exponent z g (11), namely [8] m mg g � � � � � � � � * 1 1 , with m m dz s * � . (14) This can be read as an alternative to Eq. (10) prediction for the stretching exponent g g m m m � " * . (15) The experimental data on structural relaxation shown in Fig. 1 provide evidence for mz * � 0 (13), i.e., the growth of correlations in collective dynamics under cooling. This view is based on assumption of that correlated regions of finite size RT and of a certain structure exist, for which dR /dTT � 0, near Tg , ensues a stabilization of the mate- rial-weakly-dependent parameter mz * . Such a kind of Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 807 0.8 0.7 0.6 0.5 0.4 g 3/4 PG Liquids 3BP PC toluene1/2 and fused salts GeO2 Networks SiO2 B O2 3 1/2 PIB PVC Ge As Se10 10 80 Ge As Se13 13 74 0 20 40 60 80 100 120 140 mg 0.8 0.7 0.6 0.5 0.4 g 0 50 150100 250 0.9 0.3 0,4 200 mg Polymers Fig. 1. Non-Debye against Non-Arrhenius behavior near Tg . Symbols are experimental data on dielectric, mechanical and light-scatterer relaxation, Ref. 14. Solid lines correspond to Eq. (15). The fitting parameters m m ( ) ( )pol netw� � �70 5 are found for polymeric and network glass-forming liquids, and m ( )liq � �100 10 for simple, complex and alcoholic SCLs, ex- tended by molten salts (shown in inset). The thick dashed line is given by the overall linear phenomenological fitting mg g� �250 320 , reported in Ref. 14, and the thin dashed lines indicate, approximately, the upper and lower limits of the data. model-independent clusters is observed indirectly in Fig. 1, through the parameter m * , establishing the effec- tive dimension d s (14) derived below. 3. Link to thermal expansion Let us re-present the dynamic scaling law (4) in the form R R /T a DT a /zg� ( )� � 1 . When the VFT Eq. ( 1) is ad- ditionally used for �T in Eq. (12 ), the cluster mean size R R D z T T T T VFT a g g T g c ( ) exp ,� � � � � � � � # (16) is made explicit. Furthermore, if for the fractal clusters (5) one introduces the thermal expansion coefficient $T V dV dT � 1 (17) the estimate $ T VFT f g g g g g c d z m m m T T T T T T ( ) * * ln ( ) ,� � � � � 10 2 0 0 2 (18) immediately follows from Eqs. (17) and (16). The strongly material-dependent expansion coefficient (18) is presented with the help of the known relation D m m mg g g g� � �* ( ) ln2 1 10, earlier established [14] for the VFT phenomenological form. Then, the material-in- dependent quantity \| \| ln$ g g z fT m d� 10 (19) can be readily established for fractal clusters. It is de- duced from Eqs. (17) and (13), taken at Tg , without re- course to any fitting form, including the VFT case (16). Moreover, one can examine that Eqs. (18) and (19) are self-consistent. In addition, the model equation \| \| ln(mod) $ g gT m u� 10 (20) can be obtained, if the material-independent relations m m /dz s � (14 ) and u d /df s� (8 ) are employed. A direct observation of the cluster-dimension growth (11) through Eq. (13) becomes possible thanks to the neu- tron scattering data [22,23] on the dynamical exponent z g available for supercooled polymers and analyzed in Ref. 8. Combining the result mz ( )pol � �22 2, obtained through analysis given in Fig. 2, with the output of dielec- tric and mechanical dynamical experiments for m ( )pol � � �70 5, derived in Fig. 1, one finds d s ( ) . .pol � �3 2 0 3, for the effective relaxation-space dimension (6 ). Furthermore, taking into consideration the fact of the applicability of the model finding u g ( ) . .pol � �0 33 0 03 (8), the fractal dimension (6) d f .( ) .pol � �100 0 09 is derived here for tangled-chained structures common for glass forming polymers. Moreover, a model prediction \| \| ln( ) (exp )$ % g g g g g T u m pred � � 10 1 1 (21) is obtained through Eqs. (20) and ( 14). With the aim of testing the material dependence of the model quantity de- scribed in Eq. (20), the model-equivalent relation (21) is plotted in the inset in Fig. 2. When one uses the well-known relation for the characteristic temperatures (see, e.g., Eq. (6) in Ref. 13) T T m m m g g g g0 � � (22) a new prediction follows from Eq. (21 ), namely \| \| ln ( ) ( ) (exp ) * $ % g g g g g T u m m pred 0 10 1 1 � � � . (23) This can be compared with the model-equivalent relation \| \| ln( ) (exp ) * * $ g g g g T u m m m pred 0 10 1� � � � � � � � , (24) which is smoothed through the parameter m * , obtained above in Fig. 1. Finally, our findings for the cluster-shape parameter u g (exp )(8) enables one to provide analysis for all studied glass formers, presented in Fig. 3. 808 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 V.B. Kokshenev 50 100 150 200 250 0 2 4 6 8 10 12 zg PVC T , Kg d = 1.00 0.09f � PIB PPO PVME PH 150 250 350 450 40 60 PVME PVC PBD17PBD50 PIB PPG PPG-1 PMA PVA PS PBD80 mg m = 22 2z �* Fig. 2. Cluster-dimension dynamical exponent against fragil- ity. The points are the quasi-elastic neutron scattering and fra- gility data reported for polymers in [22–24]. The solid, upper and lower pointed lines are drawn through Eq. (13) with mz* � 22 , 20 and 24, respectively. Insert: model prediction for the expansivity relation in polymers against glass transforma- tion temperature Tg . The thin and thick squares are the experi- mental data reported in, respectively, [14,22–24], re-estimated through Eq. (21) with ug (exp ) .� 033. Lines are guide to the eye. As seen in Fig. 3, the predicted expansion coefficient $ g is expected to be strongly material dependent. Con- versely, the quantity \| \|( )$ g Tpred 0 changes weakly with chemical structure in glass forming liquids and polymers. This allows to find the minimum expansivity as $ g /T(min) ( )� � �54 5 0, obtained in the fragile-glass limit ( )m /mg g & � . 4. Cluster relaxation mechanisms The mechanisms of structural relaxation in glass forming polymers and structural disordered orientational-glass formers were discussed respectively in Refs. 25 and 8. In both cases the directed by random walk (DRW) mecha- nism, known from the restricted-diffusion models for d s �1 extended over the dimension d s � 3, was estab- lished. The DRW model prediction is [8] z g DRW g ( ) � � � � � � � � 3 1 1 , (25) that can be compared with Eq. (11). It seems reasonable to extend this mechanism to the site-disordered solid o-p-H2 (OPH) mixtures, as the quadrupolar-glass (QG) former, though no dielectric loss data can be available [26]. In what follows, we seek to provide the data on the standard set of VFT and KWW dynamical exponents through the coarse-graining of microscopic model descriptions of OPH mixtures given in Refs. 27 and 28. 4.1. The case of OPH Microscopic treatment of the QG state in real OPH mixtures is based on the random-bond and random-site quadrupolar Hamiltonian introduced from the first princi- ples [26,29]. The microscopic theory was developed in Ref. 27 within the framework of Bogolyubov’s varia- tional scheme developed in terms of the two local-order dynamical variables ' i and (i . A description of the metastable rotational state, arising from the quenched site-substitutional disorder, is introduced through a set of macroscopic order parameters q x p xT i i C T i i C( ) , ( )� � " � � � � �' ( ' (2 2 2 2 , and ' 'T i Cx( ) �� , (26) given at a fixed temperature T and a rotor-molecule con- centration x corresponding to the ortho-hydrogen over- tion. Symbol C denotes a configurational average over random realizations. The microscopic treatment provides a closed system for the order-parameter equations [27]. Their analysis at high temperatures indicates that the dy- namical freezing into the short-ranged, bond-bond corre- lated quantum QG state occurs at a certain glass freezing temperature T x T xf g( ) ( )� established in both thermody- namic and dynamic NMR experiments [26]. Below Tg , the isotropic (IQG, q pT T T� �� ' ) and anisotropic ( )q pT T T� �� ' quadrupolar glass states are possible [27]. Within context of the theory of diluted magnetics, the outcomes of the QG theory are described through the two competing parameters J x J ij C /( ) � � �2 1 2 and )T i C i C x h h h h x T h x T ( ) ( ) ( , ) ( , ) � � � � � � �1 2 2 1 , (27) which are, respectively, the variance of the random ex- change J ij and the ratio of the variance h2 and mean h1 for the fluctuation field hi , both are the energetic parameters of the OPH Hamiltonian [27]. The IQG ground state ( ( ) , ,q q x0 0 01 0� � �' q p0 0�� ) defined [30] through the QG order parameter q xT ( ) estimated at T � 0 reads as q x q x q x x 0 0 2 0 2 0 2 2 1 2 1 1 3 4 1 ( ) ( ) ( ) [ ( )] max max � � " � � � � � � ) ) ) � � , )0 118( ) . maxx x x n xa � � , (28) Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 809 \| \|T$g 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 polymers PBD17PBD29PBD50 PBD80 PVC PSPVA PMA PVME PPG networks · · Ge As Sex y 1– x – ySiO2· ···· B O2 3 (N O) (SiO )2 x 2 1– x ··· · AgI PIB 40 50 60 70 80 90 100 110 120 0 20 40 60 80 molecular liquids toluene TNB mg sorbitol OTP PG 3BP glycerol · \| \|T$g 0 mg Fig. 3. Model predictions for the expansivity relations in dif- ferent glass formers against fragility. Symbols and lines corre- spond to those shown in Fig. 1. The points are dynamic relax- ation data [14] re-estimated through Eq. (23), with ug (exp ) . .,� 0 33 0 26 and 0 25. for, respectively, polymers (open squares), networks (closed circles), and molecular liquids (open circles). The solid lines are drawn through Eq. (24), with m .� 65 70 and 100 for, respectively, networks, polymers, and molecular (complex, simple, and alcoholic) liquids. where the parameter qmax establishes a certain scale of variation of q0 at the maximum concentration x max; na �12 is the number of nearest neighbors in HCP lattice. The ground-state prediction (28) was carefully tested by the experimental data (see Fig. 1 in Ref. 30 and Fig. 3 in Ref. 28). The low-T asymptotic behavior [27] q x q x q x T J x q T J x T ( ) ( ) ( ) ( ) ( ) � � � � � � � " � � � � � 0 02 2 04 4 , with J x n xa( ) � * , (29) is presented by the Taylor series and the quadrupolar cou- pling constant * � 0 82. K. Fractal clusters. Following Ref. 30, q x0( ) (28) can be improved by including distant rotor neighbors n x0( ). In this way, a fractal structure of the IQG cluster is intro- duced here through the relations n x n R R R x R x x a a d a a /df 0 0 0 1 ( ) , ( )� � � �� , for d d x xf a� �, , (30) where the fractal dimension d f which is less of the spatial dimension d � 3; R0 plays the role of the random-walk cluster correlation length, which exceeds the near- est-neighbor distance Ra . In the approximation of con- tinuos medium, x a was estimated [30] as 3 2 0 17% +, - . corresponding to the observed lower-bound critical con- centration x min (exp ) .- 0 1. Taking into account the up- per-boundary data [26] x max (exp ) .� 0 55, the modified struc- tural-disorder parameter ~ ( ) . max)0 0 118x x x n x � � , n x n x x xa a d /df 0 0 55( ) , max .� � � �� (31) is employed to fit the data [31] on q x 0 (exp ) ( ) through the modified Eq. (28). As the result, the fractal dimension d f � �2 5 0 3. . is derived via the fitting analysis (shown below in Fig. 4). Remarkably, the given coarse-grained QG description is consistent with the model-independent critical dimension d /f � 5 2, known for d � 3 in the gen- eral percolation theory (see, e.g., Table III in Ref. 21). We therefore put d ds � � 3 in Eq. (11) that justifies the appli- cation of Eq. (25) to the case of OPH. Characteristic temperatures. The orientational-order freezing mechanism rationalized [32] in terms of the short-range ordering standard molecular field competing with strongly correlated intrinsic fluctuation field. In this way, T xg ( ) is treated as a molecular-fluctuation crossover field temperature. In the weakly rotor-correlated para-ro- tational phase (PR, T � Tg ), a formation of the moder- ately supercooled state was shown [32] to be driven by the random Zeeman-type field [26]. Below Tg , the Zeeman-field effects are suppressed by the by reaction Onsager field [27,30], which, along with the molecular exchange-coupling field, determines the formation of QG-type clusters. A microscopic-level description of the so-called PR–QG boundary [26,30], here determined as T xg ( ), re- mains an unsolved problem. Nevertheless, a peculiarity in the temperature behavior of q xT ( ) near Tg can be ex- pected, when the PR–QG boundary is introduced within the framework of the coarse-grained description ~ ( )q xT , given through Eqs. (29), (28), and (31), as a crossover be- tween the moderately supercooled state (~qg g� ' ) and the strongly moderately supercooled state (~qg g� ' 2 ). In Ref. 32, this peculiarity was discussed through the two approximate schemes called by the «intrinsic–field self–compensation effect» near T xg ( ). The first scheme 810 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 V.B. Kokshenev q(x,0) 0.1 0.2 0.3 0.4 0.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 x A F R T , Kc T (x)g T (x)0 ab b a T (x)e HR PR q(x,T )g q(x,T )0 x q(x,T) Fig. 4. The quadrupolar-glass orientational-order parameter in OPH molecular mixtures against of ortho-hydrogen concentra- tion at distinct temperatures. The open circles correspond to the linear extrapolation to T � 0, after Meyer and Washburn [31]. The closed circles are given by the linear (a) and non-linear (b) extrapolations, after Sullivan and co-workers [33]. Lines: ~( , )q x 0 is the extrapolated QG ground state ~ ( )q x0 described in Eqs. (28) and (31), with d f � 2 5. and qM � 071. . The data ~( , )q x T0 and ~( , )q x Tg are Eqs.(29), modified through Eq. (31), shown at the characteristic temperatures T xg a( )( ) (32) and T x0( ) (36). Inset: the quadrupolar-glass transformation characteristic tempera- tures against concentration. The solid lines are T xg a( )( ) and T x0( ). The dashed-dotted line sketches the ergodic-instability temperature [17]. Notations: PR–para-rotational short-range or- dered (supercooled) phase, AFR — antiferro-rotational long-range ordered phase, HR — hindered rotor phase [34]. treats the crossover point as x-independent kink observed for ~ ( ) ~ ( )q x q x /T � 0 3 at the temperature T x J q q g a( )( ) ~ ~� 2 3 0 02 , (32) where the random-mean-square exchange energy J x( ) is defined in Eq. (27) and specified in Eq. (29). An alterna- tive approach considers the crossover line as an inflection point . . �2 2 0~ ( )q x / TT observed at the temperature T x J q q g b( )( ) ~ ~� 02 04 . (33) Numerical analysis of the proposed cluster description in OPH is given in Fig. 4. Thermodynamic instability. Within the mesoscopic QG-cluster treatment of orientationally-correlated rotors, given in the concentration domain x x xmin max� � , the corresponding correlation length R0 is constrained by R R Ra a� �0 3 . This implies that the distant rotors with R Ra� 3 are almost isolated relaxing units. Correspon- dingly, the rotational heat capacity C rot attributed to the ro- tational degrees of freedom of the system is commonly due to the contributions from the strongly correlated (collective) rotational excitations (Ccor ) and from the hindered rotation of weakly correlated rotors (C hind ), namely C C Crot cor hind� " . (34) The last term was described [32] through the Schot- tky-type anomaly of an isolated rotor modified by distant rotors. Similarly to the order parameter q xT ( ), the micro- scopic theory suggests the low-T asymptote for C x Tcor ( , ) represented here as C x T xR s x T T x cor ( , ) ( ) ( ) � � � �� 2 10 0 2 , (35) with R is the gas constant, T x J x s x / s x0 0 023( ) ( ) ( ) ( )� and the parameters s x0( ) and s x02( ) are given in Eq. (20) in Ref. 27. The thermodynamic-instability in Eq. (35) es- tablishes the VFT temperature T x n x q q q q 0 0 0 0 2 0 2 0 0 06 1 1 8 1 2 1 ( ) ( ~ ) ~ ~ ~ ( ~ ) ( ~ ) � � � � � � * ) ) (36) given through the functions ~ ( )q x0 and ~ ( ))0 x , n x0( ) shown in Eqs. (28) and (31), respectively, and is plotted in the inset in Fig. 4. Fragility and stretching exponent. In the absence of the loss dielectric data, the non-Debye primary relaxation in OPH mixtures could be derived from the order-parame- ter temporal behavior observed in the NMR spectra [26]. The expected VFT-type behavior, introduced here through T x0( ) and T xg ( ) suggests the OPH fragility m x m T x T x g g g ( ) ( ) ( ) � � � � � � � � � � � 1 0 1 , (37) defined with the help of Eq. (22). This finding provides the desired stretching exponent of the KWW form (2), namely g g x m m x m ( ) ( ) * � " (38) when Eq. (15) is additionally employed. Numerically, these predictions are analyzed in Fig. 5. In Fig. 5, both the versions discussed in Eqs. (36) and (32) are drawn as two ways of the order-parameter zero-temperature extrapolation. The numerical discrep- ancy between the two versions, estimated through the ex- perimental data [33], is shown by the error bar ab in Fig. 4. 4.2. Other glass formers In Fig. 6 two distinct mechanisms are suggested for site-disordered and mixed crystals and also for site-or- dered «bond-frustrated» plastic crystals, all characterized in Table 1 in Ref. 8, now extended by OPH mixtures. The first type of materials is described by the DRW model in Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 811 20 30 40 50 0.1 0.2 0.3 0.4 0.5 0.6 0.35 0.40 0.45 0.50 0.55 x a b b a mg g (o-H ) (p-H )2 x 2 1– x 0.1 0.2 0.3 0.4 0.5 0.6 x Fig. 5. Primary relaxation characteristics of OPH mixtures against ortho-hydrogen concentration: the fragility (upper plot) and stretching exponent (lower plot). The solid lines are driven through Eqs. (37) and (38) with mg * �15 for the fragility and m * � 21for the stretching exponent, common for site-disor- dered crystals [8], and T x0( ) and T xg a( )( ) are given in Eqs. (36) and (32) with the parameters found in Fig. 4. The dashed lines correspond to the case T xg b( )( ) (33). Eq. (25). Remarkably, that the same relaxation mechanism with d s � 3 is attributed to polymers [8] and also to non-poly- meric SCLs, for which though d s � 6 was found [13]. Although Eq. (14) was analyzed in Ref. 8 and m PC ( ) � 60 was found for plastic crystals, the relaxation mechanism was not identified. Adopting the Brownian diffusion ( )zcr � 2 ) as the critical regime for any subdiffusion dy- namics ( cr � 3 5/ shown in Fig. 5), the dynamic exponent (11) for plastic crystals z g CTRW g g ( ) and� � � � � � � � �6 2 1 3 5 , (39) with d g PC( ) � 6 (along with mz PC( ) �10) fits well the expe- rimentally observed data. Even though the suggested re- laxation regime in known [] only in the effective space d CTRW( ) �1 of the Continues Time Random Walk (CTRW) model, as shown in Eq. (85) in Ref. 17, Eq. (39) extends this mechanism over d CTRW( ) � 6. 5. Summary We have seen that from the macroscopic point of view no conceptual gap exists between the supercooled states in metallic and non-metallic spin glasses, dipolar and quadrupolar orientational glasses, and molecular and polymeric structural glasses. A fruitful analogy between all three fields is widely explored by many researchers [3–7] that challenges the development of a generalized theoretical consideration. In our study, a cooperative process of glass formation it treated in terms of material-ab- stract relaxing units, whose relaxation dynamics is driven by late-time correlations associated with large clusters. It is shown that the universal (material-independent) fea- tures of the $-relaxation under cooling are stipulated by the slow growing of correlations as well as by self-simi- larity of the mesoscopic-scale hierarchical structure of these correlations. Though a specification of correlations depends on the chosen theoretical scheme, their structure similarity is evidently manifested through the existence of weakly material-dependent parameters, which in turn provide a link between the dynamic exponents and ther- modynamic parameters of glass formers. As additionally shown in Ref. 17, the large clusters attributed to the late-time spatial correlations and described here through the KWW asymptotic scaling form are self-consistent with small clusters, revealed in turn through the short-time von Schweidler scaling form. As the results, this addi- tionally ensures the existence of the wide intermediate scale implicit in the universal Dixon–Nagel curve. Within this context, the typical cluster radius size RT emerges as the upper and lower bound for self-similar asymptotically small and large clusters. It has been earlier argued [8] that regardless of under- lying microscopic realizations in distinct materials, the structural relaxation is driven by local random fields in glass formers with structural disorder (including poly- mers) can be described on the mesoscopic-scale level by DRW model with d s � 3. Although the QG is the first rep- resentative of orientational glass in site-disordered crys- tals found by Sullivan’s group through the NMR spectros- copy [26], the macroscopic parameters of the standard VFT and KWW forms were not yet established. In a cer- tain sense, we fill this gap making predictions in Fig. 5. Moreover, it is shown that the QG clusters in OPH mix- tures are of fractal dimension d /f � 5 2 and they relax in space d /f � 5 2, similar to all other site-disordered glass forming materials. Remarkably, that the same relaxation mechanism is established for all simple, complex, and al- coholic liquids, though in this case d s � 6 and d f � 3 [13]. In contrast, the orientational-order relaxation in site-or- dered plastic crystals is suggested to be driven according to the CTRW model treated in space with d s � 6. Finally, the found expansion thermal expansion at Tg challenges new experimental research in SCLs, polymers and networks. Acknowledgments. The financial support by CNPq is acknowledged. 1. W. G�tze and L. Sj�gen, Rep. Prog. Phys. 55, 241 (1992). 2. P.G. Debenedetti and F.H. Stillinger, Nature 410, 259 (2001). 3. C.A. Angell, Science 267, 1924 (1995). 4. J. Souletie, J. Phys. France 51, 883 (1990). 812 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 V.B. Kokshenev g 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 (o-H ) (p-H )2 0.2 2 0.8 (BP) (BPI)0.4 0.6 zg bcr = 3/5 K Li TaO0.967 0.003 3 (KBr) (KCN)0.47 0.53 Eu Sr S0.4 0.6 site-disordered solids c-hexanol CNA c-octanol DRW CTRW plastic crystals (o-H ) (p-H )2 0.5 2 0.5 Fig. 6. Theoretical predictions for the primary relaxation mechanisms in orientational glasses: diffusion exponent against stretching exponent. The solid lines are given by Eqs. (25) and (39). Symbols (circles and squares) are drawn through the same equations and shown for materials with the known data on g (exp ) [14]. The crosses correspond to the OPH predictions on g ( )pred found in Fig. 5. 5. R.V. Chamberlin, Phys. Rev. Lett. 82, 2520 (1999). 6. M. M�zard and G. Parisi, Phys. Rev. Lett. 82, 747 (1999). 7. R.H. Colby, Phys. Rev. E61, 1783 (2000). 8. V.B. Kokshenev and N.S. Sullivan, J. Low Temp. Phys. 122, 221 (2001). 9. C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, and S.W. Martin, J. Appl. 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Colmenero, Physica A201, 38 (1993). 24. K.L. Ngai, in: Disorder Effects on Relaxation Processes: Glasses, Polymers, Proteins, R. Richert, and A. Blumen (eds.), Springer, Berlin (1994). 25. V.B. Kokshenev and N.S. Sullivan, Phys. Lett. A208, 97 (2001). 26. B. Harris and H. Meyer, Canad. J. Phys. 63, 3 (1985). 27. V.B. Kokshenev, J. Low Temp.Phys. 104, 1 (1996). 28. V.B. Kokshenev, J. Low Temp. Phys. 104, 25 (1996). 29. V.B. Kokshenev, Solid State Commun. 55, 143 (1985); V.B. Kokshenev and A.A. Litvin, Fiz. Nizk. Temp. 13, 339 (1987) [Sov. J. Low. Temp. Phys. 13, 195 (1987)]. 30. V.B. Kokshenev, Phys. Rev. B53, 2191 (1996). 31. H. Meyer and S. Washburn, J. Low Temp. Phys. 57, 31 (1984). 32. V.B. Kokshenev, J. Low Temp. Phys. 111, 489 (1998). 33. D. Zhou, C.M. Edwards, and N.S. Sullivan, Solid State Commun. 60, 901 (1986). 34. K. Kim and N.S. Sullivan, J. Low Temp. Phys. 114, 173 (1999). Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 813

Cluster relaxation dynamics in liquids and solids near the glass-transformation temperature

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