Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов

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its position at time t₀, a diffusing entity travels a kind of random walk over an average distance as a function of time. For low‐viscosity liquids and high temperatures, D is high so that entities explore a great many different positions and configurations in a time shorter than that needed to perform a physical measurement. They do it through degrees of freedom that include not only thermal motions of translation, rotation, and vibration but also the complex kinds of atomic motions collectively termed configurational, which are governed by strong short‐range repulsions and long‐range attractions in molecular liquids. The measurement then averages out all these configurations.

      Picturing these motions at a microscopic scale is difficult, however, especially for complex liquids or melts with various interacting entities. In various types of glass‐forming liquids [5], local order can nonetheless be described in terms of degree of polymerization, formation of channels or sub‐lattices, or formation of interpenetrating networks. Like the advancement of a chemical reaction, such structural features may be described in terms of the aforementioned parameter ξ. In internal thermodynamic equilibrium, i.e. in the liquid state, ξ is equal to ξ_eq(T,P), but not in the glass transition range where ξ(t) becomes a function of T(t), P(t), and A(t), revealing its nonequilibrium nature. Below the glass transition range, where the relaxation time of the configurational degrees of freedom exceeds the experimental timescale, they cease to contribute to the measured property. At temperature low enough, the structure then eventually freezes in for good in one state defined by one particular value of ξ(t), which becomes independent of the external parameters T and P.

From a practical standpoint, the timescale defined by the viscosity of the material is important to determine the temperature at which the system will fall out of equilibrium when observed at the timescale of a particular experiment. There is not yet a unique model for describing relaxation phenomena in all glass‐forming liquids (Chapter 3.7), whether strong or fragile with Arrhenian or non‐Arrhenian viscosities, respectively [6]. In measurements of macroscopic properties, one nonetheless considers generally that experimental timescales τ_exp are of the order of τ_exp~10² – 10³ seconds. The viscosity should then be of the order of 10¹² Pa.s or 10¹³ P for structural relaxation to be complete under these conditions. To stress the usually tremendous variations of viscosity down to the glass transition, it will suffice to note that the viscosities of stable liquids (i.e. above the melting or liquidus temperature) range from 10⁻³ to 10² Pa.s depending on chemical composition and structural type.

4 Glass as a Nonequilibrium Substance

Time‐dependent effects appearing at the glass transition are clearly observed in the heat capacities measured for PVAc (Figure 2), which is a model polymeric system extensively studied because of its excellent glass‐forming ability and standard T_g close to room temperature. The observed hysteresis loop between cooling and heating demonstrates that the heat capacity does not only depend on T and P but also on time. Moreover, upon heating, the heat capacity shows a typical overshoot, i.e. an endothermic event, named structural recovery process. To come back to the initial liquid state, the system needs to recover the amount of internal enthalpy that has previously been lost. From such measurements, it is possible to determine the configurational contribution to the heat capacity . Here, it is defined by the difference at every temperature between the heat capacities actually measured and estimated for the glass phase:

(6)

      This type of definition also applies to other thermodynamic variables such as the thermal expansion coefficient α_P, or the isothermal compressibility κ_T. A configurational contribution consequently represents the thermodynamic contribution that originates in configurational changes in the liquid.

      The glassy state then is defined as that for which the configurational movements have been frozen‐in, i.e. . In this state, only the vibrational motions, i.e. the fast degrees of freedom (faster than the experimental timescale), contribute. To define this contribution over the entire temperature interval of interest, an extrapolation of the glass heat capacity from low to high temperatures is needed (Figure 2). The heat capacity of the supercooled liquid can also be extrapolated toward low temperatures (Figure 2). The difference between these values for the supercooled liquid and the glass,

Figure 2 Heat capacity of PVAc measured across the glass transition range by differential scanning calorimetry at the same rate of 1.2 °C/min first upon cooling (solid circle) and then upon heating (empty circle). Dashed lines: fits made from the heat capacities measured for the glass and supercooled liquid.

(7)

then yields the equilibrium configurational contribution, which keeps increasing below T_g even though the actually observed values do vanish (Figure 3).

      From the equilibrium and actual configurational contributions, the variation of the configurational enthalpy ∆H ^conf and entropy ∆S ^conf, taken between two temperatures, are calculated with:

(8)

      where T₁ = 360 K is in Figure 2 an arbitrarily selected reference temperature.

Absolute values of both state functions could be obtained from the enthalpy and entropy of an isochemical crystalline compound through the crystallization values of these functions (see Figure 1). For lack of such a compound for PVAc, only relative values are thus presented (Figure 4) in such a way that both the actual and equilibrium values are equal from 360 K to the temperature of about 315 K at which internal equilibrium is lost. Since these variations are similar for the configurational enthalpy and entropy, only the former is shown in Figure 4.

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