First-kind phase transition

Below the T0 temperature fullerite has simple cubic lattice (set), above this temperature it has face-centered cubic lattice (feel) and in the area of T0 temperature the first-kind phase transition from sc phase to fee phase occurs. The orientation ordering takes place in fullerite that was experimentally studied in papers [6-11] and at 260 K the fee lattice was formed from the simple cubic one due to this orientation ordering. The orientation ordering is defined not only by temperature, but by pressure as well [12]. 

The potential like that shown in Figures 1.40 and 2.34b is a premise for the first-kind phase transition to occur in substance (condensation-evaporation, cry.stallizationmelting). 

Negative value of g4 < 0 corresponds to the first-kind phase transition, what will be shown further. Coefficient g2 is zero at the temperature o. The thermodynamic potential G has only two symmetrical minima at the temperatures < o. This situation deseribes the ferroelectric phase (Fig. 5.10). Above the temperature o, the third local minimum in the coordinate system origin iPs = 0) appears. This minimum is related to the paraelectric phase, which is metastable as long as the temperature reaches the phase transition temperature c. 

Fig. 5.10 Thermodynamic potential G — Go as a function of P for the first-kind phase transition for the temperatures i < o <...

Fig. 5.11 Temperature dependence of the spontaneous polarization Ps for the first-kind phase transition...

Thermodynamic potential G has only one global minimum for P = 0 above the phase transition temperature c- This global minimum might be accompanied by two local minima, which corresponds to the metastable ferroelectric phase. The spontaneous polarization jumps to zero at the Curie temperature c from the value given in Eq. (5.47) (see Fig. 5.11). Crystal energy changes also discontinuously at this temperamre, which must be accompanied by non-zero phase transition latent heat. Such phase transition is called the first-kind phase transition because of this latent heat. 

Note that the Fauler-Hugenheim isotherms take into account the first kind phase transitions in eeich layer. 

The boundary layers, or interphases as they are also called, form the mesophase with properties different from those of the bulk matrix and result from the long-range effects of the solid phase on the ambient matrix regions. Even for low-molecular liquids the effects of this kind spread to liquid layers as thick as tens or hundreds or Angstrom [57, 58], As a result the liquid layers at interphases acquire properties different from properties in the bulk, e.g., higher shear strength, modified thermophysical characteristics, etc. [58, 59], The transition from the properties prevalent in the boundary layers to those in the bulk may be sharp enough and very similar in a way to the first-order phase transition [59]. 

Phase transitions of the system such as chain ordering transitions of lipids, appear in the isotherm as regions of constant pressure in the case of first order phase transitions involving the coexistence of two phases, or as a kink in the isotherm corresponding to a second order phase transition. These kinds of surface measurements are highly sensitive to impurities and must be carried out using very pure water and sample materials. 

The experimental results discussed pertain to foam and emulsion bilayers formed of surfactants of different kinds and provide information about quantities and effects measurable in different ways. It is worth noting that analysing the observed effect of temperature on the rupture of foam bilayers enables the adsorption isotherm of the surfactant vacancies in them to be calculated. This isotherm shows a first-order phase transition of the vacancy gas into a condensed phase of vacancies, which substantiates the basic prerequisites of the theory of bilayer rupture by hole nucleation. 

The reasoning given above applies to most kinds of phase transition, for instance also for p—>a in the same system. An overview (not exhaustive) is given in Table 14.1 it also specifies the equilibrium temperature and the transition enthalpy. It should be added that it here concerns so-called first-order (phase) transitions. In section 16.1 the difference between first- and second-order transitions will be discussed. 

Thirty years later, the first Fe(ll) spin-crossover complex, [Fe(phen)2X2] (phen = 1, 10-phenanthroline, X = NCS or NCSe), was discovered by Baker et al. [6]. In [Fe(phen)2(NCS)2], the Fe(ll) atom Is surrounded by six N atoms of phen and NCS ligand molecules, and the complex exhibits a first-order phase transition associated with an abrupt LS ( A g, S = 0)-HS ( T2g, S = 2) transition at I76K, where a small thermal hysteresis attributed to the spin transition was observed. Since the discovery of the spin-crossover phase transition for [Fe(phen)2X2] (X = NCS, NCSe), various kinds of spin-crossover complexes have been found for the electron configurations of 3d" (n = 4 — 7). Most of them are Fe(ll)... 

Percolation transition is one kind of phase transitions (or critical phenomena). Unlike the melting or evaporation phase transition phenomena, which are second-order phase transitions, the percolation transition is a first-order phase transition without involving the temperature and volume changes in the system. It can be universally expressed as a power law or scaling law as shown below ... 

SSB in first-order phase transitions gas-liquid and liquid-solid are within those in the thermodynamic limit repeatedly discussed earlier and in the next section. The full micromechanisms of these phase transitions are beyond the scope of this chapter what we intend to show here is that the SSB in these transitions is triggered by the same kind of degeneracies as in the JTE and PJTE [10, 67]. 

For the exact solution of A -electron atoms at the large dimension limit, the symmetry breaking is shown to be a first-order phase transition. For the special case of two-electron atoms, the first-order transition shows a triple point where three phases with different symmetry exist. Treatment of the Hartree-Fock solution reveals a different kind of symmetry breaking where a second-order phase transition exists for N — 2. The Hartree-Fock two-electron atoms in weak external electric field exhibit a critical point with mean-field critical exponents ( = j, a = Odis, 5 = 3, and y — 1). ... 

The second transition to be discussed is a kind of lower dimensional melting. The description of the melting of a solid as a first-order phase transition is a consequence of the discontinuous change in bulk quantities at the transition point. However, every crystal is finite, and bounded by its own surface area where the process of melting may actually be initiated If there were a layer of liquid at the surface, at temperatures below the bulk melting fransition, then there is little need to activate the melting proc-... 

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