Specific heat of non-equilibrium systems and glass transition

Journal of Non-Crystalline Solids 307–310 (2002) 407–411 www.elsevier.com/locate/jnoncrysol Specific heat of non-equilibrium systems and glass transition Takashi Odagaki *, Toshiaki Tao, Akira Yoshimori Department of Physics, Kyushu University, Fukuoka 812-8581, Japan Abstract A general framework is presented to calculate the specific heat of non-equilibrium systems described by the energylandscape picture. The framework is applied to observe the time dependence of the specific heat of a model two valley system. It is shown that the glass transition can be understood as a transition from an annealed to a quenched system and that the glass transition temperature becomes lower when the observation time is increased. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 64.60.+a; 05.70.Ln; 05.40.Fb 1. Introduction There have been three different models proposed to describe the glass transition. The mode coupling theory (MCT), a mean field approach to the liquid dynamics, has been applied to super cooled liquids [1] and it has been proposed that the glass transition can be understood as an ergodicto-non-ergodic transition. Recently Parisi and coworkers [2] analyzed the static properties of systems with energy landscape, and concluded that the Kauzmann temperature at which the excess entropy vanishes is an ideal glass transition point. It is, therefore, believed that the real glass transition point where the specific heat shows an anomaly exists between these two extremes, and that the transition must be a dynamic one in sys- * Corresponding author. Fax: +81-92 642 2553. E-mail address: [email protected] (T. Odagaki). tems with energy landscape. In fact, Angell [3] gave a qualitative understanding of the glass transition on the basis of the energy landscape. However, it is still a theoretical challenge to develop a framework for obtaining the dynamic and thermodynamic quantities for systems described by the landscape. The trapping diffusion model [4] is an attempt along this line to give a unified understanding of glass transition singularities on the basis of the stochastic dynamics of a single particle and it has been shown that a dynamic transition occurs at the temperature where the mean waiting time of jumps diverges. In this paper, we give for the first time a general framework to calculate the specific heat of systems described by the energy-landscape picture as a function of the observation time and show that the abrupt change in the specific heat near the glass transition point can be understood as a transition from an annealed to a quenched system. As a simple example, we apply the framework to a two valley composed of Einstein oscillators, and show 0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 5 0 1 - 6 408 T. Odagaki et al. / Journal of Non-Crystalline Solids 307–310 (2002) 407–411 that the specific heat really exhibits a sharp change at the temperature where the structural relaxation time exceeds the observation time and that the glass transition temperature is lower for longer observation time in agreement with experiments. We also discuss the relation between the trapping diffusion model and the energy landscape picture. X oP ða; tji; 0Þ X ¼ Wab P ðb; tji; 0Þ  Wba P ða; tji; 0Þ; ot b6¼a b6¼a 2. Landscape description 3. Specific heat When the dynamics of a system is separated into slow and fast modes, we can view the system being in energy landscape with many basins, namely the fast motions occur in a basin of the landscape and the slow motion appears as jump motion among the valleys. When this description is valid, physical quantities observed at time t are expressed as an average over the probability distribution function of each basin [5]. For a quantity (type A) which depends on the initial and final valley, the average is given by In the energy-landscape picture, it is reasonable to assume that the energy of each basin is well defined. Then the energy of the system at time t is given by X hEðtÞi ¼ Ef P ðf; tÞ; ð4Þ hAðtÞi ¼ X i Pi0 X Af i P ðf; tji; 0Þ; ð1Þ f where P ðf; tji; 0Þ is the conditional probability that the system is in basin f at time t when it started basin i at time t ¼ 0 and Pi0 is the probability that the system is in basin i at t ¼ 0. For a quantity (type B) which is determined by the final valley alone the average is given by hBðtÞi ¼ X Bf P ðf; tÞ: X Pi0 P ðf; tji; 0Þ ð2Þ f Here, P ðf; tÞ ¼ i is the probability that the system is in basin f at time t. For example, the mean square displacement of a specified particle is a type A quantity, whereas the energy of the system is regarded as a type B quantity. The probability function P ðf; tji; 0Þ can be assumed to obey the master equation ð3Þ where Wab is the transition rate from basin b to basin a. The probability function P ðf; tÞ also obeys the same equation. f where Ef is the energy of basin f, and the specific heat can be formally expressed as [6]  X  dEf dP ðf; tÞ P ðf; tÞ þ Ef CV ¼ : ð5Þ dT dT f Here T is the temperature. Note the temperature dependence of the probability distribution gives an additional contribution to the specific heat. It is also interesting to note that Eq. (5) yields, in principle, a transition from the quenched expression for t ¼ 0 and to the annealed expression for t ¼ 1, though the detailed expression depends on the temperature control of the measurement. The energy of each valley is determined by the dynamics of the system in the valley. For example, if the dynamics in a valley are given by a set of oscillators whose density of states in basin f is denoted as Df ðxÞ, then Z
hx
hx coth Df ðxÞ dx: ð6Þ Ef ðT Þ ¼ 2 2kB T Here kB is the Boltzmann constant. It should be emphasized that the probability distribution P ðf; tÞ depends on every details of the temperature control for the preparation of the measurement and thus one has to specify how the measurement is performed to calculate the specific heat. 409 T. Odagaki et al. / Journal of Non-Crystalline Solids 307–310 (2002) 407–411 In order to test the validity of the framework and to obtain insight for the glass transition singularity, we introduce two simplifications: We first assume the simplified temperature control that the system is in equilibrium at temperature T at t ¼ 0 and the measurement is started at t ¼ 0. Namely we assume that the temperature T^ðtÞat time t is given by  T ðt < 0Þ; T^ðtÞ ¼ ð7Þ Ta ð0 6 tÞ: We define the specific heat by CV ¼ lim Ta !T hEðtÞi  hEð0Þi : Ta  T ð8Þ Second simplification is to consider only two valley (denoted as 1 and 2) and thus the master equation reads as P_ ð1; tÞ ¼ W21 ðT^ÞP ð1; tÞ þ W12 ðT^ÞP ð2; tÞ; ð9Þ P_ ð2; tÞ ¼ W12 ðT^ÞP ð2; tÞ þ W21 ðT^ÞP ð1; tÞ: ð10Þ We can solve this master equation readily to find for a ¼ 1 and 2   P ða; tobs Þ ¼ Paeq ðTa Þ þ Paeq ðT Þ  Paeq ðTa Þ eKðTa Þtobs ; ð11Þ where Paeq ðT Þ denotes the distribution in equilibrium and KðTa Þ ¼ W12 þ W21 : Using Eqs. (8) and (11), we find that the specific heat at t ¼ tobs is given by X CV ðT ; tobs Þ ¼ Paeq ðT ÞCa ðT Þ a  dP eq ðT Þ Ea ðT Þ 1  eKðT Þtobs þ a dT  Fig. 1. The specific heat of a two valley system composed of two Einstein oscillators with frequencies xE and 3xE . The solid curves represent (q) the quenched system tobs ¼ 0 and (a) the annealed system tobs ¼ 1. The dotted curves correspond to (1) Ctobs ¼ 102 , (2) Ctobs ¼ 104 and (3) Ctobs ¼ 106 , where exp ða=kB ðT  T0 ÞÞ is employed as the time scale of jump motion in Eqs. (13) and (14). In this plot, we set a ¼ 3
hxE and kB T0 =
hxE ¼ 0:1. CV ðT ; 1Þ ¼ X Paeq ðT ÞCa ðT Þ þ a  dPaeq ðT Þ Ea ðT Þ : dT Fig. 1 shows the temperature dependence of the specific heat for tobs ¼ 0 and tobs ¼ 1 where we employed the density of state of Einstein oscillators whose frequencies are xE and 3xE and assumed the following transition rates: W21 ¼ Cea=kB T Z2 ðT Þ ; Z1 ðT Þ W12 ¼ Cea=kB T : ð13Þ ð14Þ Here, Za ðT Þ is the partition function of valley a, C is a constant and a 3
hxE is an activation energy introduced to make the model more realistic. : ð12Þ As mentioned above, the specific heat reduces to the quenched value when tobs ¼ 0 X Paeq ðT ÞCa ðT Þ; CV ðT ; 0Þ ¼ a and when tobs ¼ 1, it becomes the annealed one 4. Degree of annealing and glass transition In real systems, the observation time should be compared with the structural relaxation time s0 which is known to behave as s0 exp ðT0 =T  T0 Þ. Here, T0 is the Vogel–Fulcher temperature. In order to incorporate the effect of diverging structural relaxation time, we replace the Boltzmann factor 410 T. Odagaki et al. / Journal of Non-Crystalline Solids 307–310 (2002) 407–411 Fig. 2. Temperature dependence of the degree of annealing for three observation times shown by the dotted curves in Fig. 1: (1) Ctobs ¼ 102 , (2) Ctobs ¼ 104 and (3) Ctobs ¼ 106 . in Eqs. (13) and (14) by exp ða=kB ðT  T0 ÞÞ. We show the specific heat thus determined by the three dotted curves in Fig. 1 for Ctobs ¼ 102 , 104 and 106 where we set kB T0 =
hxE ¼ 0:1. It is clear from Fig. 1 that the system shows a rapid decrease in the specific heat near the point where the relaxation time exceeds the observation time, and thus this point can be identified as the glass transition point. To quantify this change, we introduce the degree of annealing S defined by SðT ; tobs Þ ¼ CV ðT ; tobs Þ  CV ðT ; 0Þ : CV ðT ; 1Þ  CV ðT ; 0Þ ð15Þ Note that S ¼ 1 for the annealed system and S ¼ 0 for the quenched system. Fig. 2 shows the temperature dependence of the degree of annealing corresponding to the three observations shown in Fig. 1. It is obvious that the degree of annealing shows a rapid change near the glass transition point and can serve as an order parameter of the glass transition. 5. Results time. When the observation time is short compared to the relaxation time, the system appears to be a quenched system and when the observation time is long enough, the observed specific heat becomes annealed one which has a maximum. This maximum in the specific heat is the consequence of the stochastic motion among the basins, that is it is due to the temperature dependence of the distribution function among the valley. The observation time dependence of the specific heat can be quantified by the degree of annealing and the rapid decrease of the degree of annealing signifies the glass transition. Therefore we can regard the degree of annealing as an order parameter of the glass transition. In this framework, the glass transition temperature corresponds to the point where the structural relaxation time exceeds the observation time. Furthermore, as shown in Fig. 2, this point moves to lower temperature as the observation time is increased, in agreement with experimental observations. 6. Discussion In this paper we have presented results for simplified setting of the measurement. We can easily generalize the framework for more realistic temperature control which includes the waiting time before measurement [6]. Although our model presented here is composed of only two valley, the transition from annealed to quenched systems when the relaxation time is increased can be expected for systems with many basins. Detailed analysis for systems with many basins will be published elsewhere [7]. In passing, we discuss the relation between the trapping diffusion model [4] and the energy-landscape picture. In the trapping diffusion model, we focus on the mean square displacement of a specified particle in the system, namely 2 We have presented a general framework to calculate the specific heat of non-equlibrium systems described by the energy-landscape picture. The specific heat thus defined depends on the temperature control and thus on the observation Afi ¼ ½rn ðfÞ  rn ðiÞ is the observable, where rn ðaÞ is the position vector of nth particle in valley a. If we define the transition probability for the single particle projecting P ðf; tji; 0Þ onto the space spanned by the position T. Odagaki et al. / Journal of Non-Crystalline Solids 307–310 (2002) 407–411 of the particle, we can write the mean square displacement as X 2 hR2 ðtÞi ¼ ðs  s0 Þ pðs; tjs0 ; 0Þp0 ðs0 Þ; 411 of Education, Culture, Sports, Science and Technology. s;s0 where pðs; tjs0 ; 0Þ is that one finds the particle at s at time t when it started s0 at t ¼ 0 and p0 ðs0 Þ is the probability that the particle is at s0 at t ¼ 0. We can also derive the master equation for pðs; tjs0 ; 0Þ from the master Eq. (3) [8]. Therefore we can now understand the dynamic and thermodynamic anomalies near the glass transition point in the unified manner on the basis of the energy landscape. Acknowledgements This work was supported in part by the Grantin-Aid for Scientific Research from the Ministry References [1] W. G€ otze, in: J.P. Hansen, D. Levesque, J. Zinn-Justin (Eds.), Liquids, Freezing and the Glass Transition, North Holland, Amsterdam, 1989, p. 287; W. G€ otze, J. Phys.: Condens. Matter 11 (1999) A1. [2] M. Mezard, G. Parisi, J. Chem. Phys. 111 (1999) 1076. [3] A. Angell, Nature 393 (1998) 521. [4] T. Odagaki, Y. Hiwatari, Phys. Rev. A 41 (1990) 929; T. Odagaki, Phys. Rev. Lett. 75 (1995) 3701; T. Odagaki, Prog. Theor. Phys. Suppl. 126 (1997) 9. [5] L. Angelani, G. Parisi, G. Ruocco, G. Viliani, Phys. Rev. E 61 (2000) 1681. [6] T. Tao, A. Yoshimori, T. Odagaki, Phys. Rev. E 64 (2001) 046112-1. [7] T. Tao, A. Yoshimori, T. Odagaki, Phys. Rev. E, in press. [8] T. Odagaki, unpublished.