To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy

From general perspectives, there is no conceptual gap between the structural and orientational glasses. Both kinds of glasses expose universal features during primary relaxation (Low Temp. Phys. 33, 617 (2007)). However, in spite of much efforts made to observe generic two-level systems (TLS) in pol...

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Datum:	2009
1. Verfasser:	Kokshenev, Valery B.
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Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2009
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spelling	irk-123456789-1171092017-05-20T03:03:44Z To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy Kokshenev, Valery B. 7th International Conference on Cryocrystals and Quantum Crystals From general perspectives, there is no conceptual gap between the structural and orientational glasses. Both kinds of glasses expose universal features during primary relaxation (Low Temp. Phys. 33, 617 (2007)). However, in spite of much efforts made to observe generic two-level systems (TLS) in polymers, organic liquids, and plastic crystals via thermodynamic measurements, no similarity unifying glass formers was established. Re-analyzing a number of experimental studies, it is revealed that no renormalization conditions imposed on occupation numbers of structural units, relaxing to the glass state, were taken into consideration by the experimentalists. In this study, the effective-cluster approach is applied to configurational (excess liquid-over-solid) entropy measured in both supercooled liquids and crystals through the heat capacity. As the result, new relationships between the observable thermodynamic and dynamic characteristics are found for molecular liquids, polymers, and networks on the basis of available from the literature data. Thereby, new constraints of structure relaxation are shown to give strong evidence for the existence of TLS-type embryos of glassy structure in all studied glass formers. 2009 Article To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy / Valery B. Kokshenev // Физика низких температур. — 2009. — Т. 35, № 4. — С. 371-375. — Бібліогр.: 16 назв. — англ. 0132-6414 PACS: 61.41.+e, 61.43.Fs, 64.70.P– http://dspace.nbuv.gov.ua/handle/123456789/117109 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	7th International Conference on Cryocrystals and Quantum Crystals 7th International Conference on Cryocrystals and Quantum Crystals
spellingShingle	7th International Conference on Cryocrystals and Quantum Crystals 7th International Conference on Cryocrystals and Quantum Crystals Kokshenev, Valery B. To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy Физика низких температур
description	From general perspectives, there is no conceptual gap between the structural and orientational glasses. Both kinds of glasses expose universal features during primary relaxation (Low Temp. Phys. 33, 617 (2007)). However, in spite of much efforts made to observe generic two-level systems (TLS) in polymers, organic liquids, and plastic crystals via thermodynamic measurements, no similarity unifying glass formers was established. Re-analyzing a number of experimental studies, it is revealed that no renormalization conditions imposed on occupation numbers of structural units, relaxing to the glass state, were taken into consideration by the experimentalists. In this study, the effective-cluster approach is applied to configurational (excess liquid-over-solid) entropy measured in both supercooled liquids and crystals through the heat capacity. As the result, new relationships between the observable thermodynamic and dynamic characteristics are found for molecular liquids, polymers, and networks on the basis of available from the literature data. Thereby, new constraints of structure relaxation are shown to give strong evidence for the existence of TLS-type embryos of glassy structure in all studied glass formers.
format	Article
author	Kokshenev, Valery B.
author_facet	Kokshenev, Valery B.
author_sort	Kokshenev, Valery B.
title	To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy
title_short	To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy
title_full	To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy
title_fullStr	To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy
title_full_unstemmed	To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy
title_sort	to the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2009
topic_facet	7th International Conference on Cryocrystals and Quantum Crystals
url	http://dspace.nbuv.gov.ua/handle/123456789/117109
citation_txt	To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy / Valery B. Kokshenev // Физика низких температур. — 2009. — Т. 35, № 4. — С. 371-375. — Бібліогр.: 16 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT kokshenevvaleryb totheproblemofobservationoftwoleveltunnelingstatesinsupercooledliquidsglassformingpolymersorientationalglassesandmetallicglassesviaconfigurationalentropy
first_indexed	2025-07-08T11:39:53Z
last_indexed	2025-07-08T11:39:53Z
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fulltext	Fizika Nizkikh Temperatur, 2009, v. 35, No. 4, p. 371–375 To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy Valery B. Kokshenev Departamento de Física, Universidade Federal de Minas Gerais, Instituto de Ci �encias Exatas, Caixa Postal 702, CEP 30123-970, Belo Horizonte, Brazil E-mail: valery@fisica.ufmg.br Received January 29, 2009 From general perspectives, there is no conceptual gap between the structural and orientational glasses. Both kinds of glasses expose universal features during primary relaxation (Low Temp. Phys. 33, 617 (2007)). However, in spite of much efforts made to observe generic two-level systems (TLS) in polymers, organic liq- uids, and plastic crystals via thermodynamic measurements, no similarity unifying glass formers was estab- lished. Re-analyzing a number of experimental studies, it is revealed that no renormalization conditions im- posed on occupation numbers of structural units, relaxing to the glass state, were taken into consideration by the experimentalists. In this study, the effective-cluster approach is applied to configurational (excess liq- uid-over-solid) entropy measured in both supercooled liquids and crystals through the heat capacity. As the result, new relationships between the observable thermodynamic and dynamic characteristics are found for molecular liquids, polymers, and networks on the basis of available from the literature data. Thereby, new constraints of structure relaxation are shown to give strong evidence for the existence of TLS-type embryos of glassy structure in all studied glass formers. PACS: 61.41.+e Polymers, elastomers, and plastics; 61.43.Fs Glasses; 64.70.P– Glass transitions of specific systems. Keywords: tunneling states, structural glass transformations, supercooled liquids. 1. Introduction Structural glass transformation in supercooled liquids is one of the long-standing fundamental problems of con- densed matter physics. A process of vitrification is fol- lowed by the formation of intermediate metastable states in which a dramatic increase in viscosity and anomalous temperature behavior of transport characteristics is com- monly encountered above the glass transformation tem- perature Tg , established by scanning calorimetry. From the macroscopic point of view, no conceptual gap exists between the supercooled states in liquids and spin- (metallic and nonmetallic) glasses, orientational (dipolar and quadrupolar) glasses, or structural (molecu- lar and polymeric) glasses. As it have been recently dem- onstrated via a generalized theoretical framework [1], a cooperative process of glass formation can be treated in terms of material–abstract relaxing units, whose relax- ation dynamics is driven by late-time large-cluster spatial correlations. It has been shown that the universal (mate- rial-independent) features of the primary relaxation under cooling are stipulated by the slow growing of correla- tions, as well as by self-similarity of the mesoscopic-scale hierarchical structure of these correlations (see also Ref. 2). Though a specification of correlations depends on the chosen theoretical scheme, their structure similarity is well pronounced by the observation of weakly mate- rial-dependent parameters. In this study, the effective clusters [1] considered near the ergodic– nonergodic crossover temperature Te [3] are characterized thermody- namically by configurational entropy, attributed to the excess liquid-over-solid entropy observable via heat ca- pacity in both supercooled liquids and crystals. © Valery B. Kokshenev, 2009 Both structural glasses or orientational glasses having no long-range positional or orientational order between nonpolar or polar molecules are characterized by the for- mation of correlated metastable states. Orientational glas- ses, belonging to the class of networks within the entire family of glass formers, are divided, in turn, into two groups. One group includes structurally disordered mixed and doped molecular crystals and another one is formed by plastic crystals not affected by vibrations of the regular crystalline lattice. Similarly to supercooled liquids (trea- ted as a pattern of structural glasses), orientational glass in plastic crystals plays the role of ideal model of orien- tational glass. Below we introduce material-independent description of glass-forming materials and employ the thermodynamic models of supercooled liquids for plastic crystals. In 1972, to describe universal behavior of amorphous solids, independent from their chemical composition, An- derson with co-workers [4] and Phillips [5] proposed fa- mous tunneling two-level-system (TLS) description. Re- markably, the TLS hypothesis was anticipated by Adam and Gibbs (AG) model [6], who suggested, for metastable states in supercooled liquids, a two-state minimal-size clusters (called by them cooperatively rearranging re- gions) characterized by the minimal configurational en- tropy s ka B� ln �, with � � 2. However, in spite of much efforts made to verify the TLS hypothesis via thermody- namic measurements in glass forming polymers (with � � 3 found) [7], organic liquids (� �� ) [8], and plastic crystals (e.g., � � � for cyanoadamantane) [7] no TLS similarity unifying glasses was established. In this study we show that the experimentalists did not take into con- sideration the renormalization conditions imposed on oc- cupation numbers of structural units relaxing to the glass state. As the result, new relationships between the observ- able thermodynamic and dynamic characteristics are pro- vided for molecular liquids, polymers, and networks. 2. Background Phenomenological and model forms The phenomenological Vogel–Fulcher–Tammann (VFT) fitting form, namely � � T VFT DT T T ( ) min exp� � � � � 0 0 , (1) which also reads as log log ( ) min ( ) 10 10 0 � � T VFT VFT B T T � � � (2) with B DT� 0 10/ ln , is widely used to describe the non- Arrhenius temperature behavior of the structural relax- ation times observed in amorphous liquids and solids; D is the so-called strength index [9] and T0 is the VFT temper- ature. Proposed in the 1920s, Eq. (1) performs well within the temperature range established as T T Tg c� � [10], where Tc is the crossover temperature Tc , which sepa- rates the moderately and strongly supercooled (SCL) states [10], distinguished in the mode coupling theory [11]. The pre-factor � min (exp) � � �10 14 2 s reflects the Debye molecule vibrational times. For numerical data on the phenomenological parameters of the forms (1) and (2) es- tablished for glass-forming liquids, polymers, and orien- tational glasses, see Tables 1 and 2 in Ref. 10, and Table 1 in Ref. 12, respectively. In the AG model, the dynamic properties of SCLs are described [6] by � � � T AG AG AG T B n k T ( ) min ( ) ( ) exp� � � � � � � � � � , (3) obtained through the average transition probability 1/ T AG� ( ) of the smallest-size cooperatively rearranging regions (CRRs). Here �� ( )AG is the molar (solid-over-liquid ex- cess) chemical potential, approximated by a constant, whe- reas nT stands for the mean number of molecules which constitute the rearranging region. Equation (3) was deduced from thermodynamic consideration, given by log ( ) ( )10 � T AG T AG A C T S � � � (4) with C S n k T AG AG T B� � ( ) ( ) � � ln10 , where �S T AG( ) is the molar configurational entropy, de- fined as the excess liquid-over-solid entropy in the SCL molecular system. Adam and Gibbs treated this system as an ensemble of N /nA T of equal noninteracting clusters (NA is Avogadro’s number), which require the minimum configurational entropy s CRR min ( ) (� sa ) for the formation of a solid–liquid boundary. The total molar configurational entropy can be therefore estimated through the AG pro- posal �S s N nT AG AG A T ( ) min ( )� (5) given in Eq. (20) of Ref. 6. Furthermore, s k CRR B AG AG min ( ) min ( ) min ( ) ln ,� �� 2 (6) was suggested [6] as a first estimate for the lowest (critical) entropy, where �min is the number of distinct configura- tions available for the formation of the smallest coopera- tively rearranging region. Equations (3) and (5) provide the widely employed Eq. (4), where the AG model fitting parameters 372 Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 Valery B. Kokshenev A C N s k AG A CRR B AG� �log and min ( ) min ( ) ( ) 10 10 � � ln � (7) are commonly derived from the high-temperature data. In Ref. 13, the SCLs were studied simultaneously on the ba- sis of the dynamical data � T (exp) , derived from the dielec- tric loss spectra, and the thermodynamical experimental data on the configurational entropy, namely � � �S C T dT C C CT T T T T T T K � � �� , , ( ) ( )liq sol (8) evaluated through the excess liquid-over-solid isobaric specific heat �CT . The thermodynamic Kauzmann temper- ature TK is defined by the condition �S K � 0. Taking into consideration the experimental fact that the high-tempera- ture asymptote is observed as �C T T (exp) � �1, the AG model was specified in both thermodynamic and dynamic aspects. First, the configurational entropy (8) was found [13] in the explicit interpolation form, namely � �S S T TT K( )int � � � � � � 1 with �S C B � � . (9) Here B is the VFT-form dynamic parameter, defined in Eq. (2), while the thermodynamic parameter C is given in Eq. (4). Adopting the dynamic-thermodynamic correspon- dence for the primary timescale, we introduce here the minimum relaxing-unit sizescale na through the relations n n n n T /T T T VFT T AG a� � � � ( ) ( ) 1 0 (10) with n n k B a CRR B AG � � min ( ) ( ) ln10 �� . Here n CRR min ( ) stands for the minimum molecular size of the CRRs and n T AG( ) specifies it for the AG model, in Eq. (3). Furthermore, our consideration must be completed by the lower limit timescale � a and critical entropy sa , namely � � � min ( ) min ( ) min ( ) and VFT AG a AG as s� � � . (11) The examination made with help of Eqs. (10) and (11) re- sults in B n k C Ns k g a SL B g a B � � � �� ( ) and ln ln10 10 , (12) where additional parameters � �� g AG� ( ) and n na a SL� ( ) are introduced for the SL state. For further details, see Eq. (25) in Ref. 10. 3. How to observe the TLS A famous concept on tunneling excitations, attributed to the ground glassy state in amorphous solids [4,5], was anticipated in the AG theory [6]. Indeed, the two-level- system (TLS) hypothesis was employed in explicit form in Eqs. (5) and (6), as a lower-limit critical condition in the formation of solid-like CRRs in the normal liquid of NA molecules. The thermodynamical observation, through the heat capacity data, of the mean occupation number in the solid-like clusters n T TD( ) is commonly based on the AG proposal (5) reading as n N s ST TD A a T ( ) � � , (13) with the configurational entropy �ST estimated through Eq. (8). This can exemplified by the data [8] on CRR mo- lecular size at Tg , such as ng TD( ) . ,� 3 8 4 3. , and 4, observed through Eq. (13) in, respectively, 3-bromopentane, OTP, and toluene in the SL state, as well as by n TD glass ( ) .� 4 5, 4 6. , and 7 5. obtained through the excess entropy �S T ( )glass at- tributed to the glass state. Apart from the cluster size, Eq. (13) was used for observation of the CRR configura- tional entropy s ka B� ln min� (6), treated as the mo- del-independent fundamental parameter. Being normal- ized at the melting temperature, Eq. (13) resulted [7] in � min ( )CAD � 6, for the number of available local-axis equilibrium states in supercooled cyanoadamantane, sup- ported by its crystalline symmetry. Also, � min ( )pol � 3 and � min ( )liq �� 2 were correspondingly derived in glass-form- ing polymers [7] and expected in organic liquids [8,7]. In view of the proposed study, the following comments can be made. As was discussed in Fig. 3 in Ref. 12 and Fig. 6 in Ref. 1, plastic crystals are not dynamically con- sistent with the structurally disordered glass formers, in- cluding SCLs. Also, as independently observed in Ref. 8, plastic crystals, unlike liquids and polymers, do not obey the AG-VFT correspondence, discussed in Eq. (9). Ne- vertheless, the AG qualitative estimate (13) is not speci- fied by the structure formation of the underlying relaxing units and, therefore, it can be tested in all glass formers. Furthermore, attention should be paid to the normaliza- tion conditions. The first CRR appears below the Ar- rhenius crossover temperature T A , and thus nT � 0, in ergodic phase A [3] at T T A� , which usually lies below Tm. This means that the high-temperature asymptote is in- appropriate for the moderately cooled liquid state, that raises a question concerning the accuracy of the asymp- totic AG-VFT relation �S C/B� �(exp) (9), with all conse- quent relations given in Eqs. (12). However, this remark does not affect the experimentally observed [13] equation � �S S T T ( ) (exp)int � (9), which is now specified for To the problem of observation of two-level tunneling states Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 373 � �S n S n T T T T T a SL g c (exp) (exp) ( ) ,� � �� , (14) with the help of Eq. (10). So as evaluate the number of locally equilibrated states in strongly SCLs, associated with � min ( )SL , let us determine parameter �S � (exp) (14) through the jump in the excess specific heat �C g (exp), observed at Tg [14]. If one adopts a form � �C C T /T T g g (exp) (exp)� , compatible with the analy- sis given in Ref. 13, the useful relation � �S C m m m g g g g � � � (exp) (exp) * (15) can be deduced from Eqs. (8), (9) and Eq. (22) in Ref. 1. Taking into consideration Eqs. (14), (15) and (10), the AG proposal, given by Eq. (5) and specified in Eq. (6), reads for the ergodic supercooled liquid (SL) state as s k C R n m m m a SL B SL g g g g g ( ) min ( ) (exp) * * ln� � � � � . (16) Here R k NB A� is gas constant and the cluster molecular size ng is expected to be bounded by ng (mod) � �8 1 [10]. We have therefore shown, that the number of the energy minima on the potential energy landscape can be ob- served through the specific heat jump data, with the help of Eq. (16). Also, we offer a qualitative estimate for the absolute minimum of the configurational entropy sa in glass-like clusters, which are presumably emerge below Tg . This can be defined through the relation �S n N s ne b A a g ( )glass � , (17) which is an adaptation of the AG proposal to the tempera- ture domain T T Te g� � . The left side of Eq. (17) is given as a lower limit for the low-dense clusters, presented by the excess entropy �S e ( )glass , taken at the ergodic transi- tion temperature Te [3] and the right side is approximated by the minimum configurational entropy for CRRs formed at Tg , approximated by s s ng CRR a g ( ) � . With the help of nT (10) at Tg , and the ratio n n T Tb a SL e/ /( ) � 0, de- duced from Eq. (5) in Ref. 3, one obtains s k T T m m S R a B e g g e (min) min ( ) * ( ) ln� �� glass glass 0 (18) following from Eq. (17). For a rough numerical estimates, one adopts �S e ( )glass � � �S g (exp) laying between 20 and 25 J/K mole in SCLs [13], whose fragility number is bound by 35 � �mg 81. This yields the optimistic domain 1 5 2 3. . min ( )� �� glass , attributed to the nuclei of the glassy structure. More specific estimates provided in Table 1 follow from the auxiliary relation n n N s S T T B b g A a e c c� � � � ( )glass � 0 following from Eq. (6) in Ref. 3. This yields s k T T B S R a B c c e (min) min ( ) ( ) ln� � � � �glass glass � 0 . (19) In Table 1 the estimates are provided on the basis of Eq. (19) and dynamic and thermodynamic data available from the literature. As seen from Table 1, the TLS are well observable in supercooled liquids, instead of generally established [8] � �� . Similar to the case of Eq. (13), the estimate, pro- 374 Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 Valery B. Kokshenev Table 1. Estimation of the minimum number of local tunneling states � during formation of glassy macroscopic state Glass materials [10] mg T0, K B, K Tc , K �c �Se /[ ], ( )15 J K mol sn kB, � (19) Ortho-terphenyl (9%) 81 202 684 290 2.60 20 0.81 2.24 Salol 63 175 824 262 2.30 25 0.73 2.08 3-bromopentane 53 83 374 140 2.17 21 0.84 2.31 n-propanol 33 70 386 138 2.10 18 0.81 2.24 Methyltetrahydrofurane 65 70 407 107 2.24 23 0.57 1.76 1-propanol 31 50 806 140 2.05 16 0.44 1.55 Propylene glycol 52 122 1650 251 2.40 30 0.68 1.97 Glycerol 53 111 1431 225 2.50 23 0.54 1.71 Tri-a-naphthyl benzene 66 200 4100 424 2.20 43 0.63 1.87 Poly(ethylene terephtalate) 141 304 755 357 3.30 25 0.71 2.03 posed in Eq. (18), is not limited by SCLs. It can be applied to amorphous polymers and metallic alloys, with the help of the Te data given, respectively, in Figs. 20 and 21 in Ref. 2. These data challenge new measurements of con- figurational entropy. In vitreous network solids, the direct observation of TLS was reported in Ref. 16. 4. Conclusion We have shown how the renormalization conditions imposed on occupation numbers of structural units relax- ing to the glass state should be taken into consideration for the observation of TLS in glass forming materials. As the result, new relationships between the observable ther- modynamic and dynamic characteristics are provided for molecular liquids, polymers, and networks. The estimates made on the basis of available from the literature data give evidence for the observation of TLS-type embryos of glassy structure in all glass formers. More generally, we have shown how the earlier proposed cluster ergodic- nonergodic description [3] may provide a new tool for ac- counting for a number local-equilibrium states in Gold- stein’s energy landscape. The financial support by CNPq is acknowledged. 1. V.B. Kokshenev, Fiz. Nizk. Temp. 33, 805 (2007) [Low Temp. Phys. 33, 617 (2007)]. 2. V.B. Kokshenev, Heterostructured Molecular Clusters in Supercooled Liquids and Other Glass-Forming Materials: Dynamic and Thermodynamic Appearance in the Primary Structural Relaxation, Chapter in Atomic and Molecular Cluster Research, Y.L. Ping (ed.), Nova Science Publish- ers, New York (2006). 3. V.B. Kokshenev, Solid State Commun. 119, 429 (2001). 4. P.W. Anderson, B.I. Halperin, and C.M. Varma, Philos. Mag. 25, 1 (1972). 5. W.A. Phillips, J. Low Temp. Phys. 7, 351 (1972). 6. J.H. Gibbs and G. Adam, J. Chem. Phys. 43, 139 (1965). 7. S. Takahara, O. Yamamuro, and T. Matsuo, J. Phys. Chem. 99, 9589 (1995). 8. O. Yamamuro, M. Ishikawa, I. Tsukushi, and T. Matsuo, in: Slow Dynamics in Complex Systems, M. Tokuama and I. Oppenheim (eds.), Eighth Tohwa University Internatio- nal Symposium, AIP (1999), p. 513. 9. R. B�hmer, K.L. Ngai, C.A. Angell, and D.J. Plazek, J. Chem. Phys. 99, 4201 (1993). 10. V.B. Kokshenev, P.D. Borges, and N.S. Sullivan, J. Chem. Phys. 122, 114510 (2005). 11. W. G�tze and L. Sj�gen, Rep. Prog. Phys. 55, 241 (1992). 12. V.B. Kokshenev and N.S. Sullivan, Phys. Lett. A208, 97 (2001). 13. R. Richert and C.A. Angell, J. Chem. Phys. 108, 9016 (1998). 14. O. Yamamuro, I. Tsukushi, T. Matsuo, K. Takeda, T. Kanya, and K. Kaji, Progr. Theor. Phys. 126, Suppl., 93 (1997). 15. J.H. Magill, J. Chem. Phys. 47, 2802 (1967). 16. F. Ladieu, J. Le Cochec, P. Pari, P. Trouslard, and P. Ail- loud, Phys. Rev. Lett. 90, 205501 (2003). To the problem of observation of two-level tunneling states Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 375

To the problem of observation of two-level tunneling states in supercooled liquids, glass-forming polymers, orientational glasses, and metallic glasses via configurational entropy

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