Freezing into Solid State: Glass Formation versus

A wide variety of materials ranging from metals to polymers can solidify as glasses rather than crystals. In order to obtain solid, we can do the following experiment, that is, cooling the vapor of the material until it condenses into the liquid state, and then further gradually cooling the liquid until it solidifies.

Results of such an experiment, for a given quantity of the material, may be represented on a V (T ) plot such as the one schematically shown in Fig. 5.7.

Figure 5.7 should be read from right to left, since time runs in that di-rection during the course of the quenching (T -lowering) experiment. A sharp break or bend in V (T ) marks a change of phase occurring with decreasing T . The first occurs when the gas condenses to the liquid phase at Tb. Continued

5.4 Glass Transition 175

Fig. 5.7 The two general cooling paths by which an assembly of atoms can con-dense into the solid state. Route is the path to the crystalline state; route1  is² the rapid-quench path to the amorphous solid state.

cooling now decreases the liquid volume in a continuous fashion, the slope of the smooth V (T ) curve defining the liquid’s α = (1/V )(∂V /∂T )_P (The ex-periment is assumed to take place at P ≈ 0). Eventually, when T is brought low enough, a liquid→solid transition takes place with the exception of liquid He, which remains liquid as T → 0 under P → 0. The solid then persists in T = 0, its signature in terms of V (T ) being a small slope corresponds to the low α value which characterizes a solid. A liquid may solidify in two ways:

One is discontinuous with a crystalline solid, the other is continuous with an amorphous solid (glass).

The two solids resulting from these two quite different solidification sce-narios are labeled, correspondingly, and1  in Fig 5.7. The former occurs² at Tm. The liquid→crystal transition is marked by a discontinuity in V (T ), an abrupt contraction of the volume of the crystalline solid. But crystal-lization takes time through nucleation and growth by outward propagation of the crystal/liquid interfaces. In a quenching experiment carried out at a sufficiently low cooling rate, this is usually the route taken to arrive at the solid state. At sufficiently high cooling rates however, most materials alter their behavior and follow route  to the solid phase. T² m is bypassed without incident, and the liquid phase persists until a lower glass transition temperature Tg is reached and the second solidification scenario is realized.

The liquid→glass transition occurs in a narrow temperature interval near Tg. There is no volume discontinuity, instead V (T ) bends over to acquire the small slope characteristic of the low α of a solid.

With the liquid being cooled at a finite rate, the liquid may be taken below Tmalong the V (T ) trajectory, which smoothly continues the curve from higher T . In the temperature interval between Tmand Tg, the liquid is referred to as the undercooled liquid. If its temperature can be taken below Tgbefore

176 Chapter 5 Thermodynamics of Phase Transitions

crystallization has time to occur, the undercooled liquid solidifies as the glass and remains in this form essentially indefinitely. Glass formation, therefore, is a matter of bypassing crystallization. The channel to the crystalline state is evaded by quickly crossing the dangerous temperature regime between Tm and Tg. For a material to be prepared as an amorphous solid, cooling must proceed “fast enough and far enough”. “Far enough” means that the quenching must be taken to T < Tg, and “fast enough” implies that Tg <

T < Tm must be crossed in a time too short for crystallization to occur. In contrast to crystallization, which is heterogeneous (pockets of the solid phase appear abruptly within the liquid and then grow at its expense), the glass transition occurs homogeneously throughout the material. This transition (i.e., all liquids would form glasses) would be observed for any liquid when sufficiently undercooled and the amorphous state is a universal property of condensed matter, no matter whether ceramic, polymeric or metallic.

The fundamental difference between crystals and glasses comes from their microscopic, atomic-scale structure. In crystals, the equilibrium positions of the atoms form a translationally periodic array. The atomic positions exhibit long-range order. In amorphous solids, long-range order is absent; the array of equilibrium atomic positions is strongly disordered. For crystals, the atomic-scale structure is securely known at the outset from the results of diffraction experiments, and it provides the basis for the analysis of such properties as electronic and vibrational excitations. For amorphous solids however, the atomic-scale structure is itself one of the key mysteries.

Roughly speaking, a glass is a material that is out of equilibrium, having the disordered molecular structure of a liquid and the rigidity of a solid. An amorphous solid is metastable with respect to some crystalline phase with the thermodynamic equilibrium state of the lowest energy. While this state-ment itself is correct, the emphasis is misplaced because experience teaches that the crystalline ground state is normally kinetically inaccessible. Once formed, glasses can persist without practical limit (> 10ⁿ year). The situ-ation is similar to that of crystalline metastable diamond (the most stable crystalline structure is graphite at ambient T and P ). Since the same is true of a glass well below Tg, metastability becomes an academic matter. But the underlying physics of the glass transition remains one of the most fas-cinating open questions in materials science and condensed-matter physics.

A hotly debated issue is whether the glass transition involves an underlying thermodynamic or kinetic phase transition. The Monte Carlo simulation [14]

provides further evidence that the glass transition is not thermodynamic in origin.

A thermodynamic phase transition must involve abrupt changes in cer-tain thermodynamic properties, such as V . According to the thermodynamic viewpoint, the experimentally observable glass transition is a kinetically con-trolled manifestation of an underlying thermodynamic transition. A detailed view of the vicinity of the liquid→glass transition is shown in Fig. 5.8 for the case of the organic glass polyvinylacetate (CH2CHOOCCH3). V (T ) functions

5.4 Glass Transition 177

are cooling rate dependent where two experimental time scales are 0.02 and 100 hr while the the fixed initial temperature is well above Tg. Change this time by a factor of 5000 shift Tg by only 8 K. Thus this effect, while quite real, is small.

Fig. 5.8 V (T ) functions of an organic material with two different cooling rates in the neighborhood of the glass transition while α(T ) function is under the fast-cooling rate (0.02 hr). The slope changes in the curves signal the occurrence of the liquid→glass transition.

The reason that Tgshifts to lower T when the cooling process is extended over longer time resides in the temperature dependence of a typical molecular relaxation time tr. The quantity 1/tr characterizes the rate at which the molecular configuration of the condensed system adapts itself to a change in T . This quantity varies enormously during the cooling process. An indication of this dramatic variation is given at the top of Fig. 5.7 where in crude order-of-magnitude terms, values of trare associated with three temperatures:

Tm, Tg, and a temperature well below Tg (say, Tg-50 K). The structural-rearrangement response time may increase from the order of 10⁻¹²sec at Tm

to 10¹⁰ years at Tg-50 K. As T traverses the region near Tg, tr(T ) becomes comparable to the time scale of the measurement. As T is lowered below Tg, tr

becomes much longer than any experimentally accessible time, so that the material loses its ability to rearrange its atomic configuration in harmony with the imposed decline of T . The atoms get frozen into well defined positions (equilibrium positions, about which they oscillate), which correspond to the configuration they had at Tg: If a longer experimental time t is available, then a lower T is needed to achieve the condition tr(T ) > t, which freezes the atoms into the configuration that they maintain in the amorphous solid

178 Chapter 5 Thermodynamics of Phase Transitions

state. Note that the mildness of the t dependence on Tg is simply the other side of the coin with respect to the severity of the exceedingly steep function tr(T ).

While kinetic effects clearly play a role in the operational definition of Tg, it is generally believed that the observed glass transition is a manifestation of an underlying thermodynamic transition viewed as corresponding to the limit t→ ∞, the average cooling rate −dT/dt → 0.