Componentes clave para la integración de TIC

The Bragg-Williams model is the simplest mean-field approach applicable to ordering phenomena. The Helmholtz free energy of the alloy as a function of temperature and order parameter is:

N z

F=E-TS=Fq- — J ^ + N k J [ ( l + 0 1 n ( l + e ) + ( l - 0 1 n ( l - 0 ] , (4.2.5.1)

where Eq is the internal energy of the fully disordered state, N is the number o f sites where disordering occurs, the order parameter Q= Xb p-Xb oi=2Xa „-1, and the exchange energy J=Ej^+E^^-2E^^.

The expression (4.2.5.1) is analogous to the Landau potential (4.2.2.2), and yields the same critical exponents (e.g., near the critical point). The Bragg- Williams model has a phase transition even in the one-dimensional case; generally, as for any mean-field model, the accuracy of predictions of the Bragg-Williams model increases with the dimensionality of the system. In three dimensions this model gives qualitatively reasonable results; however, even with accurate exchange energies J, the predicted transition temperatures are usually a few times higher than the experimental ones (Redfem, 2000).

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Drawbacks of the Bragg-Williams model can be corrected by explicitly considering short-range order. In the Bethe model (see Rao & Rao, 1978), apart from the long-range order parameter Q, one or more short-range order parameters are considered. These additional parameters describe the distribution o f neighbours of both kinds in the nearest proximity of each atom. Bethe and related models yield critical exponents and transition temperatures, which are much more realistic than the mean-field predictions.

4.3. New classification of phase transitions. 4.3.1. Phase transition scenarios.

All possible modifications of a given substance correspond to minima of the free energy hypersurface. The global (i.e., the deepest) minimum is related to the phase thermodynamically stable at given p-T-conditions, while local minima correspond to metastable phases. The free energy hypersurface determines the crystal structure and many of its properties; it also determines all possible transition states, phase transition paths, and activation barriers. In principle, the free energy hypersurface is a fimction of all coordinates of all the atoms in the system (i.e., the free energy hypersurface is infinite-dimensional for an infinite system such as an ideal crystal). In principle there can be an infinite number of local minima on this surface, corresponding to an infinite number of possible metastable phases. These phases and their properties can be simulated on the computer, but it is remarkable that only a handful of them can be experimentally synthesised and studied.

The free energy hypersurface provides the most fundamental link between crystal structure, thermodynamics, and kinetics of all processes. Although the hypersurface itself is extremely complicated, we still can fruitfully exploit its concept in studying phase transitions. Often the dimensionality of the hypersurface can be effectively reduced, and only a finite part of the degrees of freedom (e.g., only those corresponding to one unit cell or a supercell) considered. Mean-field theories such as Landau theory are based on such reductions (e.g., for a system vsdth one order parameter, Landau theory reduces the infinite-dimensional hypersurface to a one­ dimensional function! - Fig. 4-1).

At this stage we would like to include metastable phases in our considerations. Let us consider the behaviour of a phase at conditions beyond its stability field.

Eventually it must transform into the state thermodynamically stable at given conditions, but at finite timescales two other scenarios are possible as well: 1) the initial phase persists as a metastable state, 2) it transforms into another metastable phase with a lower free energy, corresponding to a local minimum nearest to that of the initial phase. Numerous examples can be found for all these scenarios. Let us consider each scenario in more detail.

Transition into the stable phase. This often requires substantial structural changes and activation, in which case it can happen only at high temperatures and is catalysed by defects. If very large structural changes are required, it is likely that there will be intermediate metastable transformation products, in accordance with the famous Ostwald step transition rule. In each of these metastable states the system can stay between few picoseconds (for very shallow local minima) and indefinitely long times (for minima with high barriers). For a transition vsdth a complete structural change, many bonds should be broken in the activated state; as typical energies per bond in inorganic compounds are ~leV , such activated states become accessible only at temperatures of the order o f T - AE^/k ~ 10,000 K (!), which is well above the melting temperature of any known compound. Defects significantly lower these estimates and serve as activated centres for the nucléation of the new phase. Examples of transitions with a significant structural change include, e.g. coesite-^stishovite transition in SiOj (Fig. 4-3). Transitions between the AljSiOj polymorphs (kyanite, andalusite, sillimanite) are other examples, though in these cases some common structural elements exist in the phases (straight chains o f edge- sharing octahedral in all the AljSiOj modifications - Fig. 4-3). These transitions can occur only at elevated temperatures, being kinetically hindered at low temperatures, where metastable phases can exist indefinitely long.

The phase persists as a metastable state. Supercooled liquids (e.g., water can be supercooled) are one example. Diamond, which is metastable at ambient conditions (where graphite is stable), is another example. The three AljSiOj polymorphs can exist as metastable phases in each other’s thermodynamic stability fields - this circumstance has made the experimental determination of their phase diagram very difficult! Amorphous solids are always metastable, yet can be preserved over very long timescales. The reason why these metastable phases can be retained over practically indefinite time is that their minima on the free energy hypersurface are

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surrounded by sufficiently high energy barriers, so that the necessary activation energy can be available only at high temperatures. At sufficiently high temperatures, therefore, metastable phases inevitably transform into other phases - either stable or metastable, but with lower free energy and higher activation barriers.

Transition into another metastable state is a very interesting possibility. Such transitions are often found in nature and technology; perhaps, the clearest example is pressure-induced amorphisation (Mishima et al., 1984). When the old phase ceases to correspond to any local free energy minimum (i.e., becomes dynamically unstable), it collapses into a nearest free energy minimum; alternatively this can happen if the temperature is high enough to enable the system to jump through the activation barrier. The new metastable state must correspond to one of the local free energy minima nearest (in configurational space) to the old one. Large leaps in the configurational space are not possible, because they would require climbing high activation barriers and breaking too many bonds. This implies the maximum possible structural similarity between the old and new phases. In such transitions only those changes in structure (and, consequently, symmetry) are adopted, which are necessary to maintain the crystal’s stability against infinitesimal (dynamical stability) and finite (stability against fluctuations) atomic displacements. In the case of pressure-induced amorphisation, preservation of the dynamical stability requires the loss of translational symmetry. Even in this extreme case the local structure retains much similarity to the structure of the original crystal, the atomic displacements being rather small. More often, crystal periodicity is at least partly preserved, and changes are rather subtle, involving group-subgroup relations between symmetries of the old and new phases. In many cases it is possible to fully preserve space group.

As we have seen, symmetry is an important parameter in Landau theory, and is a natural bridge between the structure, thermodynamics and kinetics (it determines the order of the transition and the functional form of the Landau potential and possible couplings). Symmetry changes occurring upon the transition can be used to classify phase transitions.

4.3.2. New classification.

This classification is given in Table 4-1.1 divide all phase transitions into ‘global’ (where the new phase is always the thermodynamically stable phase - i.e..

corresponds to the global free energy minimum, and there are no structural relations between the old and new phases), and ‘local’ (the new phase corresponds to a nearest local free energy minimum, which implies clear structural and symmetry relations between them; the new phase can be either thermodynamically stable or metastable).

Global transitions can involve any structural and symmetry changes. These transitions are always strongly first-order and reconstructive, they require substantial activation, are often catalysed by defects, and cannot proceed at low temperatures. They occur via the nucleation-and-growth mechanism, when nuclei of the new phase are precipitated near defects or surfaces.

Local transitions, on the contrary, can be second- or higher-order, as well as first- order (often the first-order component is small). They can be reconstructive as well as displacive, order-disorder, or electronic. Often there are clear orientational relations between the old and new phases - twinning and topotaxy are common phenomena here. These transitions may require no activation and, therefore, in some cases can proceed at very low (even absolute zero) temperatures. As it was already mentioned, the new phase can be metastable. The transition path from the structure of the old phase to the new one can be described by the order parameter, specific normal mode (or modes), or ordering mechanism. According to the degree of symmetry changes, there are several possible cases (Table 4-1).

In the simplest case, there is no symmetry change upon transition. These (so- called isosymmetric) phase transitions are now being discovered experimentally in an increasing number of systems.

Next degree of symmetry change involves group-subgroup relations, while both phases are crystalline. This is the most common type of local transitions.

Further degree of symmetry change involves incommensurate modulation of the structure with partial or complete loss of translational symmetry, while preserving both short- and long-range order. Incommensurate modulation (and the corresponding loss of translational symmetry) can occur in one, two, or three dimensions.

Quasicrystals, similarly to incommensurate phases, are solids lacking crystal periodicity in one or (typically) two or three dimensions; they also have both long- and short-range order. Several types of quasicrystals are known - icosahedral quasicrystals with no lattice periodicity at all (‘3D-quasicrystals’), and quasicrystals with 1- dimensional periodicity (‘2D-quasicrystals’) having 12-fold symmetry axes

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(dodecagonal quasicrystals), 5-fold and 10-fold (decagonal), and 8-fold axes (octagonal quasicrystals). The above types of quasicrystals have symmetry axes that are incompatible with translational periodicity. There can exist ID-quasicrystals, which are simply quasiperiodic polytypes.

Table 4-1. Classification of phase transitions. See Fig. 4-3 for som e illustrations.

I. G l o b a l t r a n s i t i o n s .