Relaxation of glasses: The Kohlrausch exponent
Introduction
Since the pioneer works of Kovacs [1], [2] in 1955 and Struik [3] in 1978, the evolution of the glasses towards the equilibrium state has been studied by a great number of authors using different techniques. The relaxations below Tg are non exponential and non linear, the dependence of the relaxation time with temperature is non Arrhenius. These three “non's properties” have been discussed by several authors, references are found in the review paper of Hodge [4] and in the book of Donth [5]. Evolution of the properties of a glass (volume V, enthalpy H, creep compliance J, stress relaxation σ, dielectric constant ε, etc.) is generally described by the stretched exponential function:often called KWW (Kauzmann-William-Watt) function. n and τ are called the Kohlrausch parameters. Above Tg it is well known that this function describes the dependence of the dielectric permittivity as function of the frequency and/or time, see for example Böhmer et al. [6], [7]. The Kohlrausch relaxation time τ and exponent n are generally deduced directly from the dielectric loss spectrum. Dixon and Nagel [8] reported that n and w the width of the dielectric loss peak are inversely proportional when the temperature is changed. By numerical calculation it has been shown that the relation between n and the width w of the dielectric spectrum ε(ν) can be approximated by the relation [7], [9] :
The intersection points between the tangent at the inflexion point (t = τ) of the relaxation curve ϕ(t), and the axis ϕ(t) = 1 and ϕ(t) = 0 defines the times ti and tm called hereafter the incubation and final times. log tm / ti = log νi / νm is the width of the dielectric loss curve Tg δ versus the frequency ν. At present time there are only a few attempts for giving a physical interpretation of the Kohlrausch relaxation time, τ, and of the Kohlrausch exponent, n, below Tg. These parameters depend on the thermal history of the materials: cooling rate, pressurization rate, aging time and aging temperature etc. In this paper we discuss the origin of the Kohlrausch parameters in liquids at equilibrium and in glass.
In section 2 one recall the origin of the (Vogel–Fulcher–Tamann) VFT law observed in fragile glass formers above Tg [10], [11]. Below Tg this law is modified, taking into account that the properties of the glass depend on the structural factor; the equivalent temperature T′. This model called the generalized VFT model has been discussed in detail in ref. [12]. One gives an example, a glass is submitted to a simple T-jump. From the volume relaxation curves deduced from this model the Kohlrausch exponent is calculated. The difference between these exponents characterizing the liquid in equilibrium and out of equilibrium (Glass) is discussed.
In section 3 we show that the concept of equivalent temperature T′ (defining the incubation and final times) of the generalized VFT model explains the mechanical behavior of glassy materials after physical aging; examples of creep and stress relaxation are given.
Section snippets
Supercooled liquid at equilibrium
In 1969 Goldstein suggested that “the occurrence of secondary relaxations is an intrinsic property of the glassy state”. Goldstein and Johari [13] analyzing the dielectric loss spectra, tan δ(T) of various glass former materials, noted the presence of a secondary (Arrhenius like) transition, called β transition, 30 to 50 °C below the glass transition Tα (measured at 103 Hz). They pointed out that this transition involving simple individual motions should be considered as a precursor of the
Mechanical properties of glasses and Kohlrausch exponent
In ref. [12] we have pointed out that the concept of Deborah criteria and the application of the generalized VFT model permit to explain quantitatively the dependence of the yield stress with aging time, stain rate and temperature. We only want to show here the interest of the concept of equivalent temperature and incubation time for explaining the properties of short term creep and stress relaxation.
Conclusion
In this paper we did not discuss the possible role of free volume, order parameters, energy landscape, static and dynamic heterogeneity, etc... on the properties of glasses. We have only assumed that the glass state is a liquid out of the equilibrium which follows the generalized VFT law. In this approach of supercooled liquids, out of the equilibrium (glasses), there is no adjustable parameter. The relaxations depend on the viscoelastic parameters of the liquid at equilibrium (C1, C2, and Eβ)
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