arXiv:cond-mat/9906162v2 [cond-mat.mtrl-sci] 23 Sep 1999 Memory and chaos in the aging of a polymer glass L. Bellon, S. Ciliberto and C. Laroche Ecole Normale Supérieure de Lyon, Laboratoire de Physique , C.N.R.S. UMR5672, 46, Allée d’Italie, 69364 Lyon Cedex 07, France June 22, 2019 Abstract Low frequency dielectric measurements on plexiglass (PMMA) show that cooling and heating the sample at constant rate give an hysteretic dependence on temperature of the dielectric constant ǫ. A temporary stop of cooling produces a downward relaxation of ǫ. Two main features are observed i) when cooling is resumed ǫ goes back to the values obtained without the cooling stop (Chaos, i.e. the low temperature state is independent of the cooling history) ii) upon reheating ǫ reminds the aging history (Memory). The analogies and differences with similar experiments done in spin glasses are discussed. PACS: 75.10.Nr, 77.22Gm, 64.70Pf, 05.20−y. 1 The aging of glassy materials is a widely studied phenomenon [1, 2], which is characterized by a slow evolution of the system toward equilibrium, after a quench below the glass transition temperature Tg . In other words the properties of glassy materials depend on the time spent at a temperature smaller than Tg . In spite of the interesting experimental [3, 4, 5, 6] and theoretical progress [2, 7, 8], done in the last years, the physical mechanisms of aging are not yet fully understood. In fact on the basis of available experimental data it is very difficult to distinguish which is the most suitable theoretical approach for describing the aging processes of different materials. In order to give more insight into this problem several experimental procedures have been proposed and applied to the study of the aging of various materials, such as spin-glasses (SG)[3, 6, 9], orientational glasses (OG)[4, 10], polymers [1, 11] and supercooled liquids (SL)[5]. Among these procedures we may recall the applications of small temperature cycles to a sample during the aging time[3, 4, 5, 11]. These experiments have shown three main results in different materials: i) there is an important difference between positive and negative cycles and the details of the response to these perturbations are material dependent [3, 4, 11]; ii) for SG [3] the time spent at the higher temperature does not contribute to the aging at a lower temperature whereas for plexiglass (PMMA) [11] and OG [4] it slightly modifies the long time behavior; iii) A memory effect has been observed for negative cycles. Specifically when temperature goes back to the high temperature the system recovers its state before perturbation. In other words the time spent at low temperature does not contribute to the aging behavior at the higher temperature. These results clearly excludes models based on the activation processes over temperature independent barriers, where the time spent at high temperature would help to find easily the equilibrium state. At the same time it is difficult to decide which is the most appropriate theoretical approach to describe the response to these temperature cycles [3]. For example a recent model explains the results in SG but not in OG and in PMMA [12]. In order to have a better understanding of the free energy landscape of SG and OG, a new cooling protocol has been proposed [6] and used in several experiments [6, 9, 10]. This protocol, which is characterized by a temporary cooling stop, has revealed that in SG and in OG the low temperature state is independent of the cooling history (Chaos effect) and that these materials remind the aging history (Memory effect) [6]. The purpose of this letter is to describe an experiment where we use the new cooling protocol, proposed in ref.[6], to show that memory and chaos 2 effects are present during the aging of the dielectric constant of plexiglass (PMMA), which is a polymer glass with Tg = 388K [13, 14]. We also compare the behavior of PMMA to that of SG and OG, submitted to the same cooling protocol. To determine the dielectric constant, we measure the complex impedance of a capacitor whose dielectric is the PMMA sample. In our experiment a disk of PMMA of diameter 10cm and thickness 0.3mm is inserted between the plates of a capacitor whose vacuum capacitance is Co = 230pF .The capacitor temperature is stable within 0.1K and it may be changed from 300K to 500K. The capacitor is a component of the feedback loop of a precision voltage amplifier whose input is connected to a signal generator. We obtain the real and imaginary part of the capacitor impedance by measuring the response of the amplifier to a sinusoidal input signal. This apparatus allows us to measure the real and imaginary part of the dielectric constant ǫ = ǫ1 + i ǫ2 as a function of temperature T , frequency ν and time t. Relative variations of ǫ smaller than 10−3 can be measured in all the frequency range used in this experiment, i.e. 0.1Hz < ν < 100Hz. The following discussion will focus only on ǫ1 , because the behavior of ǫ2 leads to the same conclusions. The measurement is performed in the following way. We first reinitialize the PMMA history by heating the sample at a temperature Tmax > Tg . The sample is left at Tmax = 415K for a few hours. Then it is slowly cooled from Tmax to a temperature Tmin = 313K at the constant rate |R| = | ∂T | and ∂t heated back to Tmax at the same |R|. The dependence of ǫ1 on T obtained by cooling and heating the sample at a constant |R|, is called the reference curve ǫr . As an example of reference curve we plot in fig.1(a) ǫr , measured at 0.1Hz and at |R| = 20K/h. We see that ǫr presents a hysteresis between the cooling and the heating in the interval 350K < T < 405K. This hysteresis depends on the cooling and heating rates. Indeed, in fig.1(b), the difference between the heating curve (ǫrh ) and the cooling curve (ǫrc ) is plotted as a function of T for different |R|. The faster we change temperature, the bigger hysteresis we get. Furthermore the temperature of the hysteresis maximum is a few degrees above Tg , specifically at T ≈ 392K. The temperature of this maximum gets closer to Tg when the rate is decreased. We neglect for the moment the rate dependence of the hysteresis and we consider as reference curve the one, plotted in fig.1(a), which has been obtained at ν = 0.1Hz and at |R| = 20K/h. The evolution of ǫ1 can be quite 3 different from ǫr if we use the temperature cycle proposed in ref.[6]. After a cooling at R = −20K/h from Tmax to Tstop = 374K the sample is maintained at Tstop for 10h. After this time interval the sample is cooled again, at the same R, down to Tmin . Once the sample temperature reaches Tmin the sample is heated again at R = 20K/h up to Tmax . The dependence of ǫ1 as a function of T , obtained when the sample is submitted to this temperature cycle with the cooling stop at Tstop , is called the memory curve ǫm . In fig.2(a), ǫm (solid line), measured at ν = 0.1Hz, is plotted as a function of T . The dashed line corresponds to the reference curve of fig.1(a). We notice that ǫm relaxes downwards when cooling is stopped at Tstop : this corresponds to the vertical line in fig.2(a) where ǫm departs from ǫr . When cooling is resumed ǫ1 merges into ǫr for T < 340K. Thus the aging at Tstop has not influenced the result at low temperature, this behaviour has been called ”chaos effect” in ref.[6]. During the heating period the system reminds the aging at Tstop (cooling stop) and for 340K < T < 395K the evolution of ǫm is quite different from ǫr . In order to clearly see this effect we divide ǫm in the cooling part ǫmc and the heating part ǫmh . In fig.2(b) we plot the difference between ǫm and ǫr . Filled downwards arrows corresponds to cooling (ǫmc − ǫrc ) and empty upward arrows to heating (ǫmh − ǫrh ). The difference between the evolutions corresponding to different cooling procedures is now quite clear. The system reminds its previous aging history when it is reheated from Tmin . The amplitude of the memory corresponds well to the amplitude of the aging at Tstop but the temperature of the maximum is shifted a few degrees above Tstop . We checked that this temperature shift is independent of Tstop for temperatures where aging can be measured in a reasonable time (from 340K to Tg ). In contrast the amplitude of the downward relaxation at Tstop is a decreasing function of Tstop . It almost disappears for Tstop < 340K. This memory effect seems to be permanent because it does not depend on the waiting time at Tmin . Indeed we performed several experiments in which we waited till 24h at Tmin , before restarting heating, without noticing any change in the heating cycle. In contrast the amplitude and the position of the memory effect depend on R and on the measuring frequency. As an example of rate dependence, at ν = 0.1Hz and waiting time at Tstop of 10h, we plot in fig.3 the difference ǫm − ǫr as a function of T for three different rates. The faster is the rate the larger is the memory effect and the farther the temperature of its maximum is shifted above the aging temperature Tstop . Finally in fig.4 we compare two measurements done at two different frequencies, with the same |R| = 10K/h and the same waiting time of 10h at Tstop . 4 In this case the memory effect becomes larger at the lowest frequency and the positions of the maxima are at the same temperature. We can summarize the main results of the low frequency dielectric measurements on PMMA: (a) The reference curve, obtained at constant cooling and heating rate is hysteretic. This hysteresis is maximum a few degrees above Tg . (b) The hysteresis of ǫr increases with |R|.(c) Writing memory : a cooling stop produces a downward relaxation of ǫ1 . The amplitude of this downward relaxation depends on Tstop and it decreases for decreasing Tstop . It almost disappears for Tstop < 330K. (d) When cooling is resumed ǫ1 goes back to the cooling branch of the reference curve (Chaos). In other words the low temperature state is independent on the cooling history. (e) Reading memory : upon reheating ǫ1 reminds the aging history and the cooling stop (Memory). The maximum of the memory effect is obtained a few degrees above Tstop . (f) The memory effect does not depend on the waiting time at low temperature but it depends both on the cooling and heating rates [15]. The memory effect increases with |R|. These results seem to indicate that the more appropriate models to describe memory and chaos effects in the aging process are those based on a hierarchical free energy landscape, whose barriers growth when temperature is lowered [3, 6]. However the dependence of the memory effect on |R| and the independence on the waiting time at Tmin means that, at least for PMMA, the free energy landscape has to depend not only on temperature but also on |R|. Many models [3, 8, 16, 17] and numerical simulations [18, 19] do not take into account this dependence because they consider just a static temperature after a quench. In contrast point f) indicates that the cooling history is relevant too. Analogies between point a-b) for the hysteresis and point e-f) for the rate dependence of the memory effect leads to a new interpretation of hysteresis, which can be seen as the memory of aging at a temperature Tstop ≈ Tg . Indeed, in a free energy landscape model, when cooling the sample just above Tg the system is in its equilibrium phase, that is in a favorable configuration at this temperature. If this configuration is not strongly modified by aging at lower temperatures then, when heating back to Tg , the system reminds this favorable state, just as it does in the memory effect. It is interesting to discuss the analogies and the differences between this experiment and similar ones performed on SG [6, 9] and on OG [10]. It turns out that, neglecting the hysteresis of the reference curve of PMMA and of OG, the behavior of these materials is quite similar to that of SG. During 5 the heating period PMMA, SG and OG remind their aging history, although the precise way, in which history is remembered, is material dependent. Furthermore in these materials the low temperature state is independent on the cooling history, that is they present ”chaos effect”. One can estimate the temperature range δT where the material response is different from that of the reference curve because of the cooling stop. It turns out that the ratio δT /TG is roughly the same in PMMA, in SG and in OG, specifically δT /TG ≃ 0.2. The important difference between SG and PMMA is that the amplitude of the downward relaxation is a function of Tstop in PMMA and it is not in SG. Therefore double memory experiments cannot be simply realized in PMMA. As a conclusion the ”memory” and ”chaos” effects seems to be two universal features of aging whereas the hysteresis is present in PMMA and in OG but not in all kinds of spin glasses. It would be interesting to know if these effects are observed in other polymers and in supercooled liquids, and if the hysteresis interpretation in terms of a memory effects hold for other materials. As far as we know no other results are available at the moment. We acknowledge useful discussion with J. Kurchan and technical support by P. Metz and L. Renaudin. This work has been partially supported by the Région Rhône-Alpes contract “Programme Thématique : Vieillissement des matériaux amorphes” . 6 References [1] L.C. Struick, Physical aging in amorphous polymers and other materials (Elsevier, Amsterdam, 1978). [2] Spin Glasses and Random Fields, edit by A. P. Young, Series on Directions in Condensed Matter Physics Vol.12 ( World Scientific, Singapore 1998). [3] M. Lederman, R. Orbach, J.M. Hammann, M. Ocio, E. Vincent, Phys. Rev. B, 44, 7403 (1991); E. Vincent, J. P. Bouchaud, J. Hammann, F. Lefloch, Phil. Mag. B, 71, 489 (1995). ; C. Djuberg, K. Jonason, P. Nordblad, cond-mat/9810314. [4] F. Alberici, P. Doussineau, A. Levelut Europhysics Lett. 39, 329 (1997). [5] R. L. Leheny, S. R. Nagel, Phys. Rev.B 57, 5154 (1998). [6] K.Jonason, E. Vincent, J. Hamman, J. P. Bouchaud, P. Nordblad, Phys. Rev. Lett. 81, 3243 (1998). [7] J. P. Bouchaud, L.F. Cugliandolo, J. Kurchan, M. Mézard, in Spin Glasses and Random Fields, [2], and references therein. [8] M.Mézard, G. Parisi, M. A. Virasoro, in Spin Glasses Theory and Beyond, World Scientific Lecture Notes in Physics Vol.9 ( World Scientific, Singapore 1987). [9] T. Jonsson, K. Jonason, P. Nordblad, Phys. Rev. B 59, 9402 (1999); T. Jonsson, K. Jonason, P. Jonsson, P. Nordblad, Phys Rev. B 59,8770 (1999). [10] P. Doussineau, T. Lacerda-Aroso, A. Levelut, Europhys. Lett., 46, 401 (1999). [11] L. Bellon, S. Ciliberto, C. Laroche, cond-mat/9905160. [12] L. F. Cugliandolo, J. Kurchan, cond-mat/9812229 (1998) [13] N. G. McCrum, B. E. Read, G. Williams Anelastic and Dielectric Effects in Polymeric Solids,(Dover 1991) 7 [14] In a temperature interval of a few degrees around Tg the PMMA Young modulus changes of several orders of magnitude [13]. This Tg is consistent with the experimental evidence [11] that for T < Tg PMMA presents aging whereas for T > Tg PMMA relaxes to its equilibrium in less than 5h. [15] More details wil be given elsewhere. [16] D. S. Fisher, D. A. Huse, Phys. Rev. Lett. 56, 1601, (1987). [17] A. J. Bray, M. A. Moore, Phys. Rev. Lett. 58, 57 (1987). [18] E. Marinari, G. Parisi, J.J. Ruiz-Lorenzo, F. Ritort Phys. Rev. Lett. 76, 843 (1996); E. Marinari, G. Parisi J.J. Ruiz-Lorenzo, in Spin Glasses and Random Fields pp.59-98; E. Marinari, G. Parisi, J. J. Ruiz-Lorenzo Phys. Rev. B. 58, 14852 (1998). [19] W. Kob, J.L. Barrat, Phys. Rev. Lett. 78, 4581 (1997). 8 6 (a) 5.8 5.6 εr cooling 5.4 heating 5.2 5 320 340 360 380 temperature T (K) 400 0 −0.02 5 K/h 10 K/h 20 K/h εrh−εrc −0.04 −0.06 −0.08 (b) −0.1 330 340 350 360 370 380 temperature T (K) 390 400 Figure 1: (a) Evolution of ǫr at ν = 0.1Hz as a function of T . Reference curve obtained with |R| = 20K/h. (b) Hysteresis of the reference curve (difference between the heating and cooling curves ǫrh − ǫrc ) for 3 different |R| : 5K/h (△), 10K/h (◦) and 20K/h (▽). 9 6 (a) εr & εm 5.8 5.6 cooling 5.4 heating 5.2 5 320 340 360 380 temperature T (K) 400 cooling 0 −0.01 εr−εm −0.02 −0.03 −0.04 (b) heating −0.05 330 340 350 360 370 380 temperature T (K) 390 400 Figure 2: (a)Evolution of ǫ at ν = 0.1Hz as a function of T . The dashed line corresponds to the reference curve (ǫr ) of Fig.1(a). The solid bold line corresponds to a different cooling procedure : the sample is cooled, at R = −20K/h, from Tmax to Tstop = 374K, where cooling is stopped for 10h. Afterwards the sample is cooled at the same R till Tmin and then heated again at R = 20K/h till Tmax . (b) Difference between the evolution of ǫr and ǫm . Downward filled arrows correspond to cooling (ǫmc − ǫrc ) and upward empty arrows to heating (ǫmh − ǫrh ). 10 0 −0.01 εmc−εrc −0.02 −0.03 5 K/h 10 K/h 20 K/h −0.04 (a) −0.05 330 340 350 360 370 380 temperature T (K) 390 400 0 −0.01 εmh−εrh −0.02 −0.03 5 K/h 10 K/h 20 K/h −0.04 (b) −0.05 330 340 350 360 370 380 temperature T (K) 390 400 Figure 3: Dependence on the cooling and heating rate. Difference between ǫr and ǫm (aging at Tstop = 374K for 10h) measured at ν = 0.1Hz for 3 |R|. (a) Writing memory (cooling) : ǫmc − ǫrc at 5K/h (N), 10K/h (•) and 20K/h (H). (b) Reading memory (heating) : ǫmh − ǫrh at 5K/h (△), 10K/h (◦) and 20K/h (▽). 11 0 εmh−εrh −0.01 −0.02 1 Hz 0.1 Hz −0.03 −0.04 330 340 350 360 370 380 temperature T (K) 390 400 Figure 4: Dependence on frequency. Reading the memory (difference between heating curves ǫmh − ǫrh ) after a 10h stop at Tstop = 374K during cooling. The same rate of 10K/h is used but the measurement is done at different frequencies : ν = 1Hz () and ν = 0.1Hz (◦). 12