High-frequency asymptotics of dielectric permittivity in supercooled liquids: Experimental data and conclusions of mode-
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AND LIQUIDS
HighFrequency Asymptotics of Dielectric Permittivity in Supercooled Liquids: Experimental Data and Conclusions of ModeCoupling Theory M. V. Kondrin Institute for HighPressure Physics, Russian Academy of Sciences, Troitsk, Moscow, 142190 Russia email: [email protected] Received March 31, 2014
Abstract—The symptotic behavior of the correlation function predicted by Götze’s modecoupling theory agrees with experimentally observed relaxation processes and in most cases can be described by the phenom enological Cole–Davidson relationship. It is shown that the highfrequency behavior of the relaxation func tion is determined to a significant extent by processes related to the boundary frequency of propagation of elastic processes in liquids. A model is proposed that allows the relaxation processes to be described by unified approach in both normal and supercooled liquids. DOI: 10.1134/S1063776114090155
1. INTRODUCTION Götze’s modecoupling theory is one of the most successful theoretical models capable of describing particular aspects of liquid dynamics. Although this theory was originally intended to describe the dynam ics of glass formation (i.e., processes in sufficiently supercooled liquids), it is now commonly treated as a model describing liquid dynamics quite far (i.e., at higher temperatures or lower pressures) from the glass transition curve. This revision is to a considerable degree related to the comparison of theoretical predic tions and experimental data, in particular, on the number and form of relaxation processes observed in liquids. The modecoupling theory is based on the integro differential generalized Langevin equation adopted to describe relaxation processes in disordered media (see, e.g., [1]): ∞
d ( t) = – γ ( t' ) ( t – t' ) dt' + Ᏺ ( t ) dt
∫
(1)
0
with a memory function of the special type γ(t') ~ Ωδ(t) + γ(t), where 1/Ω ≈ 1 ps. This expansion takes into account the contributions from both rapid relax ation processes (with the characteristic frequency Ω coinciding in order of magnitude with the frequency of molecular collisions) and slower cooperative processes determining the main structural relaxation in liquids. Here, the author follows the quite popular presenta tion in [2, Ch. 2.3.2], while a more indepth analysis can be found in [3]. Equation (1) contains three unknown functions. Two additional relations, closing the system of equations and establishing nonlinear
coupling of the memory function to random force Ᏺ and response correlation function (t), follow from the fluctuation dissipation theorem: 2
γ [ ω ] ∼ c 1 [ ω ] + c 2 [ ω ], 2
γ [ ω ] ∼ Ᏺ [ ω ]. Here and below, notation on the Ᏺ[ω] and Ᏺ(t) types will imply the same function defined in the frequency and time domains. Despite considerable difficulties encountered in solving this system of nonlinear integral equations, Götze suggested an approximate asymptotic solution of the following type: a
–b
0 < b < 1. (2) This formula predicts the presence of an Sshaped region on
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