Linear and Non-Linear Dielectric Responses of a Model Glass Former Takashi ODAGAKI  , Hideaki KATOH, and Yasuo SARUYAMA 1 Department of Physics, Tokyo Denki University, Hatoyama, Saitama 350-0394, Japan 1 Department of Macromolecular Science and Engineering, Kyoto Institute of Technology, Kyoto 606-8585, Japan (Received February 14, 2011; accepted March 24, 2011; published online May 10, 2011) Exploiting a simple model for glass formers, we demonstrate that three characteristic temperatures, Vogel–Fulcher, glass transition and cross-over temperatures, can be determined from an analysis of the dielectric response. We define the relaxation time in three different ways and show that they diverge at different temperatures. From the analysis of linear and non-linear dielectric responses, we also show that the real part of the susceptibilities at the static limit becomes a cusp below the cross-over temperature and its curvature changes at the glass transition temperature. KEYWORDS: glass transition, characteristic temperature, relaxation time, dielectric susceptibility Much progress has been achieved in recent years in understanding the atomic mechanism of the vitrification process. There have been many observations which indicate existence of three characteristic temperatures in the process; the cross over temperature, T x , that signifies the change in atomic dynamics from the liquid-like to the solid-like, the glass transition temperature, T g , at which the specific heat shows a sharp change in the temperature dependence and the Vogel–Fulcher temperature, T 0 , at which the viscosity appears to diverge. The mode coupling theory 1,2) suggests that an ergodic to non-ergodic transition of liquid dynamics takes place at a critical temperature, T C , which is believed to be close to T x . A quasi-equilibrium approach based on the replica method 3,4) has been proposed to explain in a unified frame work the cross over temperature and the Kauzmann temperature, T K , at which the excess entropy appears to vanish. Although it is still controversial which temperatures must be explained, an effort has been made to understand three characteristic temperatures in a unified theoretical picture on the basis of the trapping diffusion model 5–7) whose foundation has recently been clarified by the free energy landscape theory of non-equilibrium statistical mechanics. 8,9) As for experiments, the existence of the characteristic temperatures has been suggested through measurements of viscosity, specific heat, entropy, dielectric response and various scattering experiments. 10) So far, however, these characteristic temperatures have been identified not by a single experiment but by different experiments. In this letter, we show that three characteristic temperatures can be determined by dielectric measurement alone. To this end, we employ a simple model for dielectric relaxation incorporated with barrier distribution compatible to the trapping diffusion model and show that the relaxation time and linear and non-linear susceptibilities can be utilized for determination of the characteristic temperatures. In order to elucidate the relation between dielectric responses and the characteristic temperatures, we consider an assembly of independent dipoles and assume that each dipole points in the direction or in the opposite direction of the external electric field. We represent states of a dipole by  ¼1. Then the energy of the dipole in state  is given by H ¼EðtÞ; ð1Þ where  is the electric dipole moment and EðtÞ is the external electric field. We treat the dynamics of a dipole by the stochastic model. Namely, the probability, Pð;tÞ, of a dipole being in state  at time t is assumed to obey a master equation @Pð;tÞ @t ¼ w ; Pð;tÞ w ; Pð;tÞ; ð2Þ where w ; 0 is the transition rate of the dipole from state  to  0 . We assume that a barrier exists between two states and employ w ; ¼ w 0 exp  Á  EðtÞ k B T   ; ð3Þ where Á is the barrier height and w 0 is the attempt frequency which serves as the scale of (time) 1 . Note that the transition rate satisfies the detailed balance so that the probability distribution approaches the equilibrium value after infinitely long time. It is easy to show that the polarization PðtÞ¼ fP ðþ1;tÞ P ð1;tÞg ð4Þ obeys @PðtÞ @t ¼ ðw ;  w ; Þðw ; þ w ; ÞPðtÞ: ð5Þ For the static electric field EðtÞ¼ E 0 (constant), we investigate the relaxation function ðtÞ¼ hPðtÞi  P eq hPð0Þi  P eq ; ð6Þ where P eq is the polarization in the equilibrium state, P eq ¼  tanh E 0 k B T   : ð7Þ In eq. (6), h  i denotes an ensemble average over the distribution of the barrier height of each dipole. When an oscillatory electric field EðtÞ¼ E 0 e i!t is applied, we observe the response of the system through the susceptibilities. We first expand the polarization PðtÞ in a power series of the external field. PðtÞ¼P 0 ðtÞþP 1 ðtÞþP 2 ðtÞþP 3 ðtÞþ : ð8Þ Here, P k ðtÞ denotes the k-th order term of the expansion, from which we define the k-th order susceptibility  k ð!Þ through  E-mail: odagaki@mail.dendai.ac.jp Journal of the Physical Society of Japan 80 (2011) 053705 053705-1 LETTERS #2011 The Physical Society of Japan DOI: 10.1143/JPSJ.80.053705