Anomalous Tunneling Systems in Amorphous Organic Materials. Abstract

Transcription

1 Anomlous Tunneling Systems in Amorphous Orgnic Mterils S. Shling, 1 M. Koláč, 2 V.L. Ktkov, 3, nd V.A. Osipov 3 1 Institut für Festkörperphysik, TU Dresden, Dresden, Germny 2 Chrles University, Prgue, Czech Republic 3 Bogoliubov Lbortory of Theoreticl Physics, rxiv: v1 [cond-mt.dis-nn] 13 Oct 2016 Joint Institute for Nucler Reserch, Dubn, Moscow region, Russi Abstrct We compre the het relese dt of orgnic glsses with tht of morphous nd glss like crystlline solids. Anomlous behvior ws found in ll these mterils, which disgrees with the stndrd tunneling model. We cn explin the most of the experimentl observtions within phenomenologicl model, where we ssume tht for prt of tunneling systems the brrier heights re strongly reduced s consequence of the locl stress produced during the cooling process. PACS numbers: Eb, B, Hk, Gk Electronic ddress: ktkov@theor.jinr.ru 1

2 I. INTRODUCTION Let us begin with n overview of the min results nd conclusions concerning the het cpcity nd het relese experiments in inorgnic nd glsslike crystlline mterils. It hs been recently shown tht brrier heights of two-level systems (TLSs) in both morphous nd glsslike crystlline solids cn be drsticlly reduced by frozen in locl mechnicl stresses cused by rpid cooling of the smple [1]. The vlue of the reduction cn be estimted in the frmework of Eyring model [2] tht is widely employed for describing stress-induced yielding nd non-liner mechnicl response in polymer glsses [3]. Strong locl fluctutions of the stress re expected to exist in glsses nd the mximum brrier height V mx is reduced to V mx fter the cooling of the smple with V mx V mx = σ mx V c where V c is so-clled ctivtion volume ( typicl volume required for moleculr sher rerrngement) nd σ is the frozen in stress cused by the therml expnsion. As consequence, the distribution of the brrier heights becomes much more complicted thn tht ssumed within the stndrd tunneling model (STM) [4, 5], which gives roughly constnt distribution P (V ) = P 0. Nmely, there ppers step in the distribution function t V mx (see Fig. 1). Moreover, the relxtion time of the tunneling process lso exhibits step t the corresponding relxtion time τ mx. In the one-phonon pproximtion, the relxtion rte of thermlly ctivted process is written s τ 1 t = A(E 2 0/k 3 B) coth(e/2k B T ). (1) One gets τ t = τ mx t 0 = min 0 where the minimum tunneling energy min 0 = (2E 0 /π) exp( V mx /E 0 ), (2) E 0 is the zero-point energy [6], nd A = 8π3 k 3 B ρh 4 ( γ 2 l υ 5 l ) + γ2 t. (3) υt 5 TLSs with the reduced brrier heights were clled nomlous (TLSs). The results gree well with the predictions of the STM for ll prmeters when the experimentl time scle t is less thn τ mx. For exmple, in vitreous silic τ mx is found to be of the order of 100 s, nd the spectrl density obtined from the het cpcity (P C ), therml conductivity, ultrsound, dmping (internl friction) gree quite well with the STM [7]. However, the het relese ws 2

3 mesured t longer time up to 10 5 s nd for t > τ mx gives spectrl density P Q roughly 4 times smller thn the spectrl density P C deduced from the het cpcity. One cn estimte the vlue of the step s (P + P n )/P n = P C /P Q, where smll contribution of nomlous TLSs to the het relese t t > τ mx (P 0 in Fig. 1) hs been neglected (explntion see below). We get P /P n = 3 for vitreous silic, which mens tht the min prt of TLSs is nomlous [7]. It should be stressed tht the most of thermodynmicl prmeters tken from experiments do not llow us to distinguish between the contribution of norml nd nomlous TLSs, while this is possible by the het relese mesurements. The het cpcity experiment yields the spectrl density P C, which is n verge vlue of ll tunneling systems between τ min nd t, where τ min is the minimum relxtion time of the TLSs nd t is the time of the het cpcity mesurement (typiclly, t 1 s). Thus, the het cpcity lwys gives the spectrl density for t < τ mx, tht is P C = P + P n. The het relese is cused by TLSs with the relxtion time close to the mesuring time t (numericl clcultions with stndrd distribution function show tht the het relese t the time t is produced by TLSs with the relxtion time t/10 < τ t < 20t). Their corresponding brrier heights re locted in quite smll rnge V ( )V m, where V m is n verge vlue, which determines the relxtion time τ t = t. Thus, the het relese experiments give us the spectrl density for well-defined brrier height V m nd, therefore, by incresing time one cn move from the left to the right side of the step in the distribution P (V ). Notice tht the new prmeter τ mx plys n importnt role in our model. Nmely, in ccordnce with Fig. 1 we get different results for both the het relese nd its correltion to the het cpcity in three different regions: 1. V m < V mx 2. V m = V mx 3. V m > V mx nd t < τ mx nd t = τ mx nd t > τ mx In the first region we reproduce the results of the STM with new spectrl density P = P + P n = P C = P Q. Explicitly, C p = (π 2 kb/12)p 2 C T ln(4t/τ min ), (4) where τ min = τ t ( 0 = E), nd dq/dt Q = (π 2 kb/24)p 2 Q V s (T1 2 T0 2 )t 1, (5) 3

4 FIG. 1: The normlized distribution function vs the reltive brrier height V/V mx : (1) corresponds to the stndrd tunneling model with constnt distribution up to the mximum brrier height V mx, (2) shows the distribution function ccording to the model of nomlous tunneling systems. The locl stress reduces the brrier heights for prt of tunneling systems (nomlous tunneling systems) nd their mximum brrier height is reduced to V mx. At V > V mx distribution function is minly determined by norml tunnelling systems, where the locl stress increses the brrier heights or the brrier heights remin unchnged (no stress). A smll mount of TLSs ppers in this rnge too due to the distribution of the locl stress (the curve 3). The rrows (4), (5) nd (6) indicte the possible rnges of effective brrier heights corresponding to the time windows of het relese mesurements. The rrow (7) corresponds to time rnge of the het cpcity mesurement. the with V s being the volume of the smple, T 1 the strting, nd T 0 the finl temperture. At low tempertures (T 0, T 1 < T c0 ), for the het relese we found no difference from the STM. Here T c0 is the crossover temperture, where the relxtion rte of the tunneling τ 1 t is equl to the relxtion rte of the thermlly ctivted process. Above the crossover temperture the relxtion time of norml TLSs decreses rpidly with incresing temperture due to 4

5 the Arrhenius lw. As consequence, prt of TLSs relxes during the cooling process nd does not contribute to the het relese. In this cse, the het relese devites from T 2 1 T 2 0 behvior. At high enough tempertures T 1 > T (T is the freezing temperture), ll TLSs with the energy E > 2.4k B T rech the stte of equilibrium during the cooling process, nd the het relese sturtes. Thus mximum vlue of the het relese will be observed t T 1 T in Eq. (5). T depends weekly on the cooling rte R t this temperture nd is directly proportionl to the brrier height V m of the TLSs cusing the het relese t given time fter the cooling to temperture T 0 < T c0. Notice tht the freezing temperture is close to the crossover temperture T n 1.2T c0 for norml TLSs (see [8]). However, the brrier height of nomlous TLSs increses with incresing temperture nd their freezing temperture T cn be essentilly higher thn T n. In this cse, gint het relese will be observed since the het relese sturtes t very high tempertures T 1 = T. In the second region, the het relese relxes very fst nd cn be described by the reltion Q = [Q l + (Q s Q l ) exp( t/τ mx )]t 1, (6) where Q l nd Q s re proportionl to P 0 +P n nd P +P n, respectively. In the het relese one cn directly see the step in the distribution function nd determine the mximum relxtion time of nomlous TLSs s function of n verge energy E v /k B = 2.4(T 1 + T 0 )/2. This rre cse ws observed in ZrCO [9]. Fig. 2 shows the het relese s function of time fter cooling from different T 1 to T 0 = 1.3 K. Notice tht Eq. (6) describes the time dependence very well. Fig. 3 shows the energy dependence of τ mx, where we point out nother interesting fct: the mximum energy is not proportionl to 1/E v, s expected from the STM (see Eq. (1)) but roughly proportionl to E v T 1 + T 0. In the region 3 there is remrkble discrepncy between P C nd P Q since P C P Q = P + P n P 0 + P n > 1. (7) In this cse, the het relese is minly determined by the norml TLSs, nd the mximum het relese is much smller thn the vlue mesured t t < τ mx. Nevertheless, there is lso smll contribution of nomlous TLSs (P 0 in Fig. 1). It ws suggested tht this smll contribution to the het relese t t > τ mx could be cused by some distribution of the locl stresses in the smple fter rpid cooling. Within this scenrio, the mximum locl stress will reduce V mx to V mx fter the cooling. For smller stress, the corresponding V will tke 5

6 FIG. 2: The het relese in 14.5 cm 3 (ZrO 2 ) 0.89 (CO) 0.11 fter cooling from different initil tempertures T 1 to the finl phonon temperture T 0 s function of time (t = 0 t the beginning of cooling) [9]. The curves re clculted with Eq. (6), which corresponds to the distribution P (τ) of the stndrd tunneling model with step t τ mx nomlous TLSs. The fit prmeter τ mx is given in Fig. 3 the vlues between V mx cused by the cut-off in the distribution of nd V mx t the relxtion time t > τ mx, nd corresponding freezing temperture T vries between T n < T < T. Thus, the distribution of the locl stresses leds to contribution of nomlous TLSs t t > τ mx. Such contribution ws observed in ll mterils with P C > P Q. Notice tht P 0 is smll in comprison to P - for exmple, P 0 /P = 0.05 for vitrous silic [7]. Nevertheless, for high initil tempertures this frction of nomlous TLSs gives the dominnt contribution to the het relese (roughly 2 times lrger thn the contribution of norml TLSs) since they hve lso much higher T 0 in comprison to T n (see Fig. 4). Mesuring the T 1 -dependence of the het relese one cn seprte the contribution of norml nd nomlous TLSs. Fig. 4 shows the prmeter Q 1 = t Q s function of T 2 1 T 2 0. The contribution of norml TLSs sturtes for -SiO 2 t T n = 5.5 K while for nomlous TLSs (P 0 ) it sturtes bove 14.1 K. Vlues of these two contributions re proportionl to P n nd P 0, correspondingly. τ mx depends on the vlues 6

7 FIG. 3: The fit prmeter τ mx (used to fit the curves of the het relese dt s function of time for different mterils (see Figs. 2,5,7-10)) s function of T 1 + T 0, which is close to n verge energy of relxing tunneling systems. of locl stresses nd V mx s well s on the mss of tunneling entity. The lst possibility ws demonstrted for TLSs in NbTi cused by H or D in [1]. The het relese nd het cpcity dt of ll investigted inorgnic morphous nd glsslike crystlline solids complies with one of these three cses (see Tble I). A trnsition between cses one nd three is rrely observed since chnging mesuring time in the het relese experiments by fctor of ten modifies the corresponding brrier height by few percents only. All three cses were observed in ZrO 2 CO [9] (see Fig. 2), PLZT [10] nd lso in -SiO 2 [7] nd NbTi-H [1]. It should be noted tht the het relese in morphous orgnic mterils hs not yet been nlysed in detil. At the sme time, the experimentl studies of orgnic glsses show rther specific behvior of the het relese. In this pper, we mke comprison of the het cpcity nd het relese dt between orgnic glsses, inorgnic glsses, nd glsslike crystlline mterils. Our nlysis clerly indictes the existence of nomlous TLSs in ll these mterils; their possible origin is the locl stresses during the cooling process. 7

8 FIG. 4: The fit prmeters Q 1 = t Q s function of T1 2 T 0 2 for different mterils (see Figs. 2,5, 7-10). A chrcteristic feture of nomlous TLSs is high freezing temperture T. II. HEAT CAPACITY AND HEAT RELEASE OF AMORPHOUS ORGANIC MA- TERIALS Our purpose is to nlyse the relevnt het relese nd het cpcity dt for orgnic morphous mterils nd show tht nomlous TLSs should lso exist in these mterils. If this is the cse, the distribution function must be similr to tht in inorgnic glsses with step t V mx. There is, however, n importnt difference: in orgnic mterils the het relese decys s t with < 1 in contrst to both the predictions of STM nd the t 1 behvior observed in most of inorgnic glsses. As possible explntion one cn ssume n dditionl strong increse of P (τ) ner τ mx. Let us consider n extreme cse of δ-like growth. In this cse, the het relese should be written s Q = Q n /t + Q (8) with Q = [Q /t + P x ] exp( t/τ mx ), (9) 8

9 FIG. 5: The het relese Q/m s function of time for PMMA. The experimentl dt re tken from [11, 12]. The curves correspond to Eq. (10). Dshed lines show the first term in Eq. (10). The fit prmeters Q 1 nd τ mx re given in Figs. 3 nd 4. where P x is time independent contribution to the het relese. Otherwise, we would expect to find for P Q vlue lrger or close to P C for t < τ mx, nd fst relxtion of the het relese t t τ mx. A. Het cpcity nd het relese of PMMA One of the first long time het relese experiments were performed by Zimmermnn nd Weber with -SiO 2 nd PMMA [11, 12]. In both mterils the poor greement with the STM ws observed. The results of vitreous silic were considered nd discussed in detil in ref. [7]. The results of PMMA re shown in Fig. 5. For low T 1, the het relese is roughly proportionl to t 1 s predicted by the STM. However, mrkedly different time dependence ws observed for higher T 1. A much better greement we get with fit Q = (Q 1 /t + P x ) exp( t/τ mx ), (10) 9

10 where Q 1 = Q + Q n. (11) The het relese of PMMA is t low tempertures T 1 nd short time minly determined by the term Q l /t nd grees here quite well with the STM. However, Eq. (10) yields much better greement with the experimentl dt including tht dt t higher T 1 nd long time. As is seen, tht Eq. (10) differs from Eqs. (8) nd (9): the exponent cuts down lso Q n, nd therefore this fit fils t t τ mx. In order to seprte terms Q nd Q n we need the het relese dt either for higher tempertures T 1 or for t τ mx. Unfortuntely, they re not yet vilble for PMMA. As we see in Fig. 4, T 2 1 T 2 0 dependence of Q 1 grees with the STM. Thus, we cn estimte the distribution prmeter by using the STM. The result is P Q = Jm 3 t ρ = 1.19 g/cm 3. Notice tht this prmeter relly mrkedly exceeds the vlue deduced from the het cpcity P C = Jm 3 [13]. This finding indictes tht the het relese is strongly influenced by the new term P x which is lso strongly proportionl to T 2 1 T 2 0 (see Fig. 6). This mens tht ll contributions to the het relese hve the sme or similr distribution function of the energy (P (E) = const). Notice tht the increse of the distribution function with V could explin the devition from t 1 behvior of the het relese. However, in this cse we would obtin devition from T 2 1 T 2 0 behvior s well. In prticulr, the model with TLS-TLS interction tken into ccount gives P 1/ 2 0 [14]. This function ws successfully used to fit the dt of spectrl diffusion in PMMA [15]. However, using this distribution function in our cse we obiin Q T 5/2 1 T 5/2 0, which is inconsistent with the het relese experimentl dt (see Fig. 4). The contribution of the nomlous TLSs nd norml TLSs cnnot be seprted becuse this requires the het relese experiments with higher T 1 which were not performed up to now. B. Het relese of 3-MP/2,3-DMB Het relese mesurements were performed with 3-methylpentne/2,3-dimethylbutn t much higher strting temperture thn for PMMA (up to 20 K) [16]. The het relese is found to be roughly proportionl to t 0.7 nd relxes much fster for very long time (see Fig. 7). Notice tht even shrp cut of the TLSs distribution function could not provoke 10

11 FIG. 6: The fit prmeters P x s function of T1 2 T 0 2 used for the clcultion of the het relese dt of PMMA nd 3-mp/2,3-dmb with Eq. (10). such strong dying in Q. The full lines present the fit curves ccording to Eq.(10). The corresponding prmeters Q 1 nd P x re shown in Figs. 4 nd 6 s function of T 2 1 T 2 0. Q 1 is evidently proportionl to T 2 1 T 2 0 for T 1 < 3K nd one cn estimte P Q = Jm 3. At higher T 1, the results devite from the liner dependence. However, no sturtion of the het relese s function of T 1 is observed up to 20 K, which is typicl behvior of nomlous TLSs. This is cler indiction tht t < τ mx. At long time we observe fster relxtion due to τ mx. The dditionl contribution P x determines the het relese fr long time. The energy dependence is similr to tht for Q 1. The prmeter τ mx is close to the corresponding vlues of PMMA with similr energy dependence for low T 1 (see Fig. 3). Interesting results were obtined for this mteril by mesuring the het relese s function of the finl temperture T 0 (see Fig. 8). The vrition in T 0 from 1.6 K to 3.2 K does not influence the prmeter τ mx. However, strong reduction of τ mx is observed for T 0 > 3.2 K. This mens tht bove 3.2 K the therml ctivtion process domintes, nd we expect sturtion of the het relese s function of T 1 round 4 K provided tht the 11

12 FIG. 7: The het relese of 21 cm 3 3-mp/2,3-dmb s function of time fter cooling from different tempertures T 1 (see the insert) to 1.6 K [16]. The relxtion becomes very rpid for long time. The curves re clculted with ccording to Eq. (10). The fit prmeters P x, Q 1 nd τ mx given in Figs. 3, 4 nd 6. re brrier heights re constnt. In fct, Q 1 is no more proportionl to T1 2 T0 2 for T 1 > 3 K (the het relese cused by norml TLSs sturtes). However, no sturtion is observed up to 20 K which is strong indiction tht the further increse of the het relese t T 1 > 4 K is cused by nomlous TLSs. C. Het cpcity nd het relese of PS Het relese mesurements in polystyrene were performed by Nittke et. l [17]. In the rnge of 0.7 h < t < 6 h t T 0 = 0.2 K with 0.5 K < T 1 < 1K mesurments show tht the het relese nd Q 1 re proportionl to t 1 nd T1 2 T0 2, respectively (see Fig. 17 in [17]). Resulting P Q = J/g, is in good greement with P C = J/g deduced from the het cpcity [17, 18]. All results re in perfect greement with the STM nd no 12

13 FIG. 8: The het relese of 21 cm 3 3-mp/2,3-dmb s function of time fter cooling from 20 K to different tempertures T 0 (see the insert) [16]. The curves correspond to Eq. (10). contributions of nomlous TLSs ws observed. Notice, however, tht the het relese ws mesured for quite short time nd low T 1 only. The contribution of nomlous TLSs would be much lrger for high T 1 nd long time. Indeed, the het relese mesured fter cooling from 80 K to 0.3 K in wide time rnge shows the typicl time dependence cused by the nomlous TLSs (see Fig. 9) with τ mx = 60 h. D. Het cpcity nd het relese of epoxy resin The het relese of epoxy resin ws mesured fter cooling from 292 K to 1.15 K during 25 dys [19]. It ws found to be proportionl to t 0.76 (see Fig. 9). A similr time dependence ws found for epoxy glue Stycst [20]. long τ mx Eq. (8) gives good fit for epoxy with very = 350 h. At lower T 1 the power lw in the time dependence does not chnge: the reltion between Q 1 nd P x is the sme for different T 1. This holds only when both contributions re cused by nomlous TLSs. Q 1 devites from the STM t T 1 > 3K (the sturtion of the het relese cused by the norml TLSs) but the het relese is still 13

14 FIG. 9: The het relese of epoxy resin [19] nd PS [17] s function of time fter cooling from very high temperture T 1 to T 0 (for tempertures see the insert). The curves correspond to (10). incresing up to the highest temperture T 1 = 22 K. This is gin cler indiction tht the het relese is produced by nomlous TLSs. The step in the distribution function ws found for higher T 0 in the thermlly ctivted rnge [19] (see Fig. 10). Here the therml bsorption ws mesured fter heting the smple from T 1 to T 0 (i.e. T 1 < T 0 ). Since the therml ctivtion lters the time dependence for shorter time s well, we fit the dt by power lw Q = P (t 0 )(t 0 /t) exp( t/τ mx ), (12) where P (t 0 ) is the het relese t t = t 0 (t 0 τ mx ) nd the fit prmeter depends on T 0. The prmeter τ mx 50 h t 2.23 K. chnges rpidly with incresing temperture T 0 from 3 h t 3.95 K to For better understnding, Fig. 11 shows these dt together with the clculted temperture dependencies within the STM. At the crossover temperture T c0 = 3 K the tunneling relxtion rte becomes equl to the rte of therml ctivtion. Thus the vlue of τ mx t the lowest temperture lies in the tunneling rnge, while two others re in the rnge of therml ctivtion. The Arrhenius lw τ 1 t = τ 1 0 exp( V/k B T ) with τ 0 = s yields effective 14

15 FIG. 10: The het bsorbed in epoxy resin fter rpid heting from T 1 to T 0 (tempertures re given in the insert) [19]. The curves correspond to Eq. (12). brrier heights V m /k B = 163 K (T 0 = 3.33 K) of the TLSs cusing t this time the het. It is expected within the STM (where brrier heights re temperture independent) tht the corresponding relxtion time decreses very rpidly with incresing temperture (see Fig. 11, dshed line). For norml tunneling systems this leds to the sturtion of the het relese t the freezing temperture T n, which is bout 20% higher thn T co, i.e. T n = 3.6 K. However, no sturtion ws found up to 22 K (see Fig. 4). In ddition, the mesured τ mx 3.95 K is 3 orders of mgnitude longer thn expected from the Arrhenius lw (open squre in Fig. 11). All these fcts demonstrte tht the effective brrier height increses t higher temperture s shown in Fig. 12. Both the brrier height nd the corresponding relxtion time in Fig. 11 were clculted with the following fit function (the curve in Fig. 12): V mx /k B = [ exp( 16.5K/T )]K. (13) We cn estimte the freezing temperture T = 60 K from the T 2 1 T 2 0 t dependence of the prmeter Q 1 in Fig. 4 by extrpoltion of the experimentl dt t high nd low T 1 ; finlly we get from Eq. (8) in [1] the brrier height V m /k B = 2800 K, which is close to V mx. One cn lso estimte the mximum brrier height from the internl friction experiments. 15

16 FIG. 11: The mximum relxtion time τ mx used in the fit curves in Fig. 10 with Eq. (12) s function of temperture. The vlue obtined t 4 K (full squres) is 3 orders of mgnitude longer thn expected from the Arrhenius lw (open squres). No mximum of the dmping ws observed in experiments with epoxy up to 100 K (f = 150 MHz) [21]. This leds to mximum brrier height lrger thn 1800 K. A dmping pek nerby 110 K cn be estimted for the TLSs in the vibrting reed experiments (f = 84 KHz) with stycst [22]. This yields V mx /k B = 2860 K (full squres in Fig. 12). All these prmeters re in resonble reltion. Thus, the TLSs with the mximum brrier heights become frozen in stte t 60 K (open circles in Figs. 11 nd 13) nd their relxtion time increses with decresing temperture, reches mximum vlue of s t bout 15 K thn returns to short time 10 4 s t 4 K (see Fig. 13). Let us note tht the Arrhenius lw gives s (!) for the relxtion time of TLSs with the mximum brrier height t 4 K. E. Het relese of pentnol-2 Pentnol is good cndidte for proton trget due to its high hydrogen content (C 5 H 11 OH). However, high polriztion requires n uniform distribution of the prmgnetic centers, which is much better in morphous thn crystlline mterils. A stndrd 16

17 FIG. 12: The effective mximum brrier height V mx clculted from the het relese (full squres) nd internl friction dt (open squres) of epoxy resin s function of temperture. The curve corresponds to fit with Eq. (13). cooling of pentnol with dilution refrigertor yields polycrystlline pentnol. A difficult procedure ws necessry to get the morphous structure. The pentnol ws mixed with 5 % wter nd cooled down rpidly by liquid nitrogen (the glss trnsition temperture T g lies ner 170 K). Then the solid pentnol ws mounted rpidly in the dilution refrigertor nd cooled down to lower tempertures. This procedure is not too convenient. Therefore we check Pentnol-2, where the OH-group is trnsferred from the end of the long molecule to the side. Indeed, the worse symmetry of the molecules leds to n morphous structure by norml cooling in dilution refrigertor. Het relese experiments showed clerly glssy behvior, however with devitions in the time dependence: the het relese ws found to be proportionl to t 0.69 between 20 nd 500 min [23]. Eq. (8) gives lso good fit of the time dependence. The corresponding prmeters Q 1 re shown in Fig. 4. The crossover temperture is ner 3 K nd one expects sturtion within the STM bove 4 K. However, the T 1 -dependence does not sturte up to 12 K. The dt re close to tht of epoxy resin (see Fig. 4). 17

18 FIG. 13: The mximum relxtion time τ mx of epoxy resin s function of temperture clculted with the temperture-dependent mximum brrier height V mx τ mx in Fig. 12 (the curve). Full squres: deduced from the het relese dt in Fig. 10, the open circle indictes the relxtion time t the freezing temperture deduced from the internl friction nd het relese dt. III. DISCUSSION We cn mke three min conclusions from the bove-reviewed experimentl dt for morphous orgnic mterils: 1. The min contribution to the het relese comes from nomlous TLSs. Indeed, the het relese sturtes t very high strting tempertures T 1 ( gint het relese), the distribution prmeter P Q is close to P C (or lrger) nd the het relese relxes fster t very long time. 2. The het relese experiments with orgnic glsses correspond to the considered cses (1) nd (2), where t τ mx. In fct, τ mx in orgnic glsses is longer thn in inorgnic ones or glsslike crystlline mterils, except NbTi/D (see Fig. 3). 3. In contrst to inorgnic mterils, for long time, but t < τ mx the het relese is 18

19 roughly proportionl to t with 0.5 < < 0.8. We lso put emphsis on n brupt decrese of the het relese with time in 3-mp/2,3-dmb, which cnnot be described even by suggesting cut in P (V ). Let us nlyse these conclusions with reltion to the model of locl mechnicl deformtions due to gint lrge-scle fluctutions in therml expnsion during the cooling of smple [24]. According to this model, the therml expnsion coefficients α(r) tke rndom vlues inside the dilttion centers of the smple. The dispersion in the distribution of α(r) is suggested to be much bigger thn its men vlue: α 2 / α 2 [D/(ΓE v )] , where D is the deformtion potentil, Γ is the Grüneisen prmeter. Rpid cooling from n initil temperture T 1 to finl T 0 results in genertion of strong locl mechnicl stresses, which provokes liner reduction in the brrier height. As consequence, the life time of high-energy TLSs reduces nd their het relese becomes experimentlly observble. Nmely these TLSs re clled nomlous. This model llows us to explin qulittively n increse of τ mx with n verge energy of TLSs E v (see Fig. 3). Indeed, the dispersion α 2 decreses when T 0 +T 1 grows. This results in growing brrier heights of nomlous TLSs, reducing of 0, nd incresing relxtion time. Regrding the second item of our conclusions, the edge in the distribution function of nomlous TLSs is shifted to the left ginst the edge of norml TLSs on the vlue V = σv c. This step cn be explined under the ssumption of fixed limit of locl stresses σ mx. Tking into ccount the structurl chrcteristics of the mterils one cn suggest tht σ mx hs mximl vlue for inorgnic crystlls, little lesser for morphous mterils nd hs smller vlue for orgnic polymers. Indeed, the corresponding inverse behvior of τ mx tkes plce (see Fig. 3). Devitions from the t 1 behvior in the het relese cn be relted to the increse of the spectrl density P 0 (t) t 1 during the cooling process. This mens grdul increse of locl stresses nd/or sizes of stress concentrted regions. In the lst cse, n increse of P 0 will be ccompnied by shift of V mx to the left. This cn explin n brupt decrese of Q(t) in 3-mp/2,3-dmb. It should be stressed tht there is no evidence of ny devitions from t 1 behvior in the het relese in inorgnic mterils. Therefore, suggested increse of locl stresses should be specific property of orgnic mterils. We suppose tht locl stresses re influenced by n expnsion of the smple during the het relese process. Indeed, n verge 19

20 therml expnsion coefficient α is relted to C p, Grüneisen prmeter Γ nd isotherml compressibility κ T s α = κ T ΓC p. (14) According to (4), the het cpcity grows with time. At the sme time, in orgnic polymers κ T nd τ min hve severl orders bigger nd lesser vlues, respectively, in comprison to the inorgnic mterils. A decrese of τ min is result of strong electron-phonon interctions in polymers. Hence in polymers we cn expect more pronounced influence of α. Even slow drift of α cn provoke formtion of dditionl locl stresses nd, s consequence, rising quntity of nomlous TLSs. As nother possible explntion of this phenomenon (which would lso pprove Eq. (8)) one cn suggest the existence of n unknown dditionl chnnel which provides the relxtion of TLSs in the time τ mx independently of the brrier heights. Unfortuntely, the origin of this sort of TLS relxtion is still n open question. A possible cndidte is the resonnt relxtion bsed on TLS-TLS interction t very low tempertures suggested in [14]. IV. SUMMARY Summrising, our nlysis shows tht ll glsses (morphous orgnic, inorgnic, nd glsslike crystlline mterils) revel n universl behvior of the reltion between the het relese nd the het cpcity in good greement with our model of nomlous TLSs (see Tble I). The het relese shows similr behvior in vrious orgnic polymers nd hs some peculir properties in comprison with inorgnic glsses nd glsslike crystls. We hve shown tht despite these peculirities ll existing het relese dt cn be explined within the model of locl mechnicl stresses where specific properties of orgnic polymers re cused by lower vlue of the mximum stress nd more pronounced drift of n verge therml expnsion coefficient due to bigger vlue of isotherml compressibility nd stronger electron-phonon interction. It should be stressed tht orgnic glsses re convenient for further experimentl investigtions of the gint het relese owing to their longer τ mx. 20

21 TABLE I: Distribution prmeters extrcted from the het relese nd the het cpcity mesurements for different mterils. Mteril P C P Q P C /P Q T co T n T 0 T cse Ref J 1 m J 1 m 3 K K K K -SiO t > τ mx [7] NbTi t > τ mx [1] NbTi 9%H t > τ mx [1] ZrO 2 CO t > τ mx [9] ZrO 2 CO > 64 t < τ mx [9] NbTi 9%D t < τ mx [1] PMMA > 2 t < τ mx [11, 12] Epoxy resin t < τ mx [19] PS > 1 t < τ mx [18] 3pm/2,3dmb >20 t < τ mx [16] pentnol >12 t < τ mx [23] This work hs been supported by the Heisenberg-Lndu Progrm under Grnt No. HLP

22 List of Symbols α Γ γ l,t 0 κ T λ ρ σ τ 0 τ The therml expnsion coefficient. Grüneisen prmeter. /2 u ik : Effective deformtion potentil for longitudinl or trnsversl phonons. The tunneling energy. Isotherml compressibility. The Gmow prmeter. Mss density. Mechnicl stress. Pre-exponentil fctor in the Arrhenius lw. Relxtion time of tunneling systems due to phonon-ssisted interction for nomlous TLSs. τ t τ t Relxtion time of tunneling systems due to phonon-ssisted interction. Thermlly ctivted relxtion time of TLSs. τ min Minimum relxtion time of TLSs. A C p D E 0 E v m P 0 P P C P n P Q P 0 A prmeter proportionl to γ 2 /ρυ 5 4 with dimensions J 3 s 1 The specific het t constnt pressure. Deformtion potentil of TLSs. The zero-point energy. Averge energy of TLSs cusing the het relese. The mss. Constnt density of sttes of TLSs Constnt density of sttes of nomlous TLSs. Constnt density of sttes deduced from the het cpcity. Constnt density of sttes of norml TLSs. Constnt density of sttes deduced from the het relese. Constnt density of sttes of nomlous TLSs deduced from the het relese long time mesurement (t > τ mx ). P x A fit prmeter of Eqs. (8) nd (10). R Cooling rte in het relese experiments. 22

23 T Freezing temperture; below it nd for typicl cooling rtes the TLSs remin in nonequilibrium stte nd contribute to the het relese. T 0 T 1 T co Mesuring temperture in het relese experiments. Chrging temperture in het relese experiments. Crossover temperture where the thermlly ctivted relxtion time equls the tunneling relxtion time. T T n Freezing temperture for nomlous TLSs. Freezing temperture for norml TLSs. Q 1 A fit prmeter of Eq. (10). Q A fit prmeter of Eq. (8). Q l A fit prmeter of Eq. (6) Q n A fit prmeter of Eq. (8). Q s A fit prmeter of Eq. (6). V V V m Potentil brrier height. Potentil brrier height of nomlous TLSs. The verge brrier height of the TLSs cusing the het relese (is directly proportionl to the freezing temperture). V s V c υ l,t Volume of the smple Activtion volume. The sound velocity. 23

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