Part 2: Entropy and Phase Transformations 34
4.6 Conclusions
∝ εn, as shown in Fig. 4.7. Exponents n were found to be 1.50, 1.52, 1.57, 1.64, 1.68, and 2.00 for the as-prepared material, 200◦C, 260◦C, 310◦C, 360◦C, and bcc Fe, respectively. This may be interpreted as a change in the dimensionality [39] from 2.5 for the as-prepared material to 2.68 in the material annealed at 360◦C. However, the APT results show no obvious trend indicating a reduced dimensionality.
Chapter 5
Changes in entropy through the glass transition in Cu-Zr
5.1 Introduction
The atomic structure and dynamics of liquids and glasses are much less understood today than the properties of crystalline solids. First-principles theories have answered many questions about crystalline materials and can even be used to make predictions of phonons, for example. These successes are owed to the ability of translational symmetry and Blochs theorem to provide detailed descriptions of crystalline solids. However, this approach fails immediately when used to describe the complex many-body interactions in liquids and glasses [62]. Thus, the nature of amorphous materials and the glass transition remains one of the most challenging problems in solid state physics [63].
The glassy state is ubiquitous in nature and technology [64]. Window glass is the most widely- known glass, composed of sand (SiO2), lime (CaCO3), and soda (Na2CO3)1. The term ‘glass’
encompasses any solid that possesses a non-crystalline structure and exhibits a glass transition when heated towards the liquid state. However, metallic glasses are quite distinct in their physical properties from other types of glasses. Metallic glasses are much tougher than oxide glasses and ceramics, and also tend to have higher tensile yield strengths and higher elastic strain limits than polycrystalline metal alloys [66].
1Many who recognize the amorphous nature of window glass also fall prey to the common myth that the liquid-like structure of window glass can be seen in the thickness variation of stained glass windows in old cathedrals. In fact, solving the Volger-Fulcher-Tamman expression for the viscosity as a function of temperature yields a relaxation time on the order of 1032years, which is well beyond the age of the universe [65].
The glass transition is unique to materials that are in an amorphous state at temperatures below the crystallization temperature of the material. Over a narrow temperature range immediately below the crystallization temperature, the solid amorphous material softens, becoming a viscous liquid that is deeply undercooled below the usual melting temperature. One of the most intriguing aspects of the glass transition is that the atomic structure of the supercooled liquid does not change significantly across the transition, while the transport properties such as viscosity change by more than ten orders of magnitude [67,68,69,70].
The glass transition can be regarded as a kinetic phenomenon in which the rapidly-increasing time scale for structural equilibration of the supercooling liquid crosses the time scale of the experimental tools used to study the material [67]. When a liquid is cooled, its molar volume, enthalpy, and entropy decrease, and there is a concomitant increase in its viscosity and relaxation time. Fig.5.1 illustrates the temperature dependence of a liquid’s volume (or enthalpy) at constant pressure. Upon cooling below the freezing point Tm, atomic motion slows down. If the liquid is cooled sufficiently fast, crystallization can be avoided. Eventually, atoms will rearrange so slowly that they cannot adequately sample configurations in the available time allowed by the cooling rate. The liquids structure therefore appears ‘frozen on the laboratory timescale (for example, minutes). This falling out of equilibrium occurs across a narrow transformation range where the characteristic atomic relaxation time becomes of the order of 100 seconds, and the rate of change of volume or enthalpy with respect to temperature decreases abruptly (but continuously) to a value comparable to that of a crystalline solid. The resulting material is a glass. The behavior depicted in Fig.5.1is not a true phase transition, as it does not agree with Ehrenfest’s requirement that a discontinuity be observed in a derivative of the Gibbs free energy with respect to some thermodynamic variable.
The observed behaviors indicate that changes inV,H, andS may be quantitatively related to changes in both the phonon properties and atomic configurational dynamics. Relations between the frequency of vibrational modes and the heat capacity are understood in terms of the Debye Einstein theory and Gr¨uneisen parameters, and these relations are usually regarded as satisfactory for most solids. It may be expected that both the free volume and Sexc of a liquid would have additional
Volume/Enthal
Tg Tm Temperature glass
crystal
Figure 5.1: Temperature dependence of a liquid’s volume or enthalpy at a constant pressure.
Tm is the melting temperature. For glasses that are cooled sufficiently quickly, the liquid enters the supercooled liquid regime before atomic motions become ‘frozen’ on the laboratory time scale, resulting in the glass with a higher volume and enthalpy and volume than its corresponding crystal.
contributions from atomic vibrations and associated anharmonic forces, and that these contributions would decrease on cooling. However, the significance of these additional contributions is generally neglected in both free volume and entropy theories. These contributions to heat capacity and entropy can be described qualitatively in terms of potential energy landscape theory [71] or inherent state model [72]. This leads to an insight into the interdependence of the configurational and vibrational contributions to Cp and entropy of an equilibrium liquid. Measurements of the total heat capacity, including both of these entropy contributions, are made routinely [73, 74]. However, a definitive measurement of either the total configurational or vibrational entropy contribution has not been made.
The configurational entropy model of Adam and Gibbs [75] hypothesizes that the progressively increasing size of the cooperatively rearranging regions and decreasing configurational entropy in a liquid as temperature decreases is responsible for the apparently diverging relaxation times and viscosity. The transition from the glassy state to the supercooled liquid state is accompanied by a positive jump in heat capacity in this narrow range of temperature defined as the glass transition
temperature.
Their theory yielded the quantitative expression
η=η◦exp(C/TSc), (5.1)
which connects the viscosityηwith the configurational entropySC, where C is a constant containing a free enthalpy barrier to cooperative rearrangements, andT is temperature.
Vibrational entropy is noticeably absent in Adam-Gibbs theory, as the excess entropy associated with the jump in heat capacity is assumed to be caused entirely by the configurational entropy.
However, this assumption has not been tested experimentally, and there is no a priori reason to believe that that this is the case. Understanding of the contribution of the vibrational entropy to the excess entropy of the supercooled liquid plays a critical role in building a quantitative description [76,77,78,79,80].