Continuous phase transitions transformation

As it is pos.sible to continuously transform liquids into ga.ses and vice versa without passing through a phase transition, Langton lumps liquid and gi s phases into tlie more general fluid phase. 

Cp is the specific heat at constant pressure, k is the compressibility at constant temperature. The conversion process of a second-order phase transition can extend over a certain temperature range. If it is linked with a change of the structure (which usually is the case), this is a continuous structural change. There is no hysteresis and no metastable phases occur. A transformation that almost proceeds in a second-order manner (very small discontinuity of volume or entropy) is sometimes called weakly first order . 

Even when complete miscibility is possible in the solid state, ordered structures will be favored at suitable compositions if the atoms have different sizes. For example copper atoms are smaller than gold atoms (radii 127.8 and 144.2 pm) copper and gold form mixed crystals of any composition, but ordered alloys are formed with the compositions AuCu and AuCu3 (Fig. 15.1). The degree of order is temperature dependent with increasing temperatures the order decreases continuously. Therefore, there is no phase transition with a well-defined transition temperature. This can be seen in the temperature dependence of the specific heat (Fig. 15.2). Because of the form of the curve, this kind of order-disorder transformation is also called a A type transformation it is observed in many solid-state transformations. 

When the free energies F of the two crystal structures are identical, the system is at a critical point. The identity of F does not imply identical fimctions (otherwise the two phases would be indistinguishable). Therefore, at the critical point first derivatives of F might differ and therefore enthalpy, volume, and entropy of the two phases would be different. These transformations are first-order phase transitions, according to Ehrenfest [105]. A discontinuous enthalpy imphes heat exchange at the transition temperature, which can easily be measured with DSC experiments. A discontinuous volume is evident under the microscope or, more precisely, with diffraction experiments on single crystals or powders. Some phase transitions are however characterized by continuous first derivatives of the free energy, whereas the second derivatives (specific heat, compressibility, or thermal expansivity, etc.) are discontinuous. These transformations are second-order transitions and are clearly softer. 

As opposed to the liquid-crystal transformation, the liquid-glass transformation is not a phase transition and therefore it can not be characterized by a certain transition temperature. Nevertheless, the term "the vitrification temperature , Tv, is widely used. It has the following physical meaning. As opposed to crystallization, vitrification occurs when the temperature changes continuously, i.e. over some temperature interval, rather than jump-wise. Inside this interval, the sample behaves as a liquid relative to some of the processes occurring in it, and as a solid relative to other processes occurring in it. The character of this behaviour is determined by the ratio between the characteristic time of the process, t, and the characteristic relaxation time of the matrix, x = t//G, where tj is the macroscopic viscosity and G is the matrix elasticity module. If t x, then the matrix should be considered as a solid relative to the process, and if t > x it should be considered as a liquid. The relation tjx = 1 can be considered as the condition of the matrix transition from the liquid to the solid (vitreous) state, and the temperature Tv at which this condition is realized as the temperature of vitrification. Evidently, Tv determined by such means will be somewhat different for the processes with different characteristic times t. However, due to the rapid (exponential) dependence of the viscosity rj on T, the dependence of Tw on t (i.e. on the kind of process) will be comparatively weak (logarith-... 

We all know that when a liquid transforms to a crystal, there is a change in order the crystal has greater order than the liquid. The symmetry also changes in such a transition the liquid has more symmetry than a crystal since the liquid remains invariant under all rotations and translations. Landau introduced the concept of an order parameter, , which is a measure of the order resulting from a phase transition. In a first-order transition (e.g., liquid-crystal), the change in is discontinuous, but in a second-order transition where the change of state is continuous, the change in is also continuous. Landau proposed that G in a second-order (or structured) phase transition is not only a function of P and T but also of and expanded G as... 

A 2D first order phase transition is unequivocally characterized by a discontinuity in the T E) isotherm at / = constant. At this special point in the r E) isotherm an expanded" 2D Meads phase is transformed into a more condensed" 2D Meads phase. Expanded and condensed Meads phases have to coexist. Therefore, 2D nucleation and growth occur in the presence of a preformed expanded but supersaturated Meads adsorbate. The surface concentration of the expanded Meads adsorbate, F, is continuously changing according to the actual polarization state of the UPD system. As long as corresponding -amounts (cf. eq. 3.4) contribute significantly to the overall measured -values, an identification of 2D nucleation and growth is not possible. 

Simulations of octahedral molecular clusters at constant temperature show two kinds of structural phase changes, a high-temperature discontinuous transformation analogous to a first-order bulk phase transition, and a lower-temperature continuous transformation, analogous to a second-order bulk phase transition. The former shows a band of temperatures within which the two phases coexist and hysteresis is likely to appear in cooling and heating cycles Fig. 10 the latter shows no evidence of coexistence of two phases. The width of the coexistence band depends on cluster size an empirical relation for that dependence has been inferred from the simulations. 

Given that (see Fig. 9.8) at the glass transition temperature, the specific volume Vs and entropy S are continuous, whereas the thermal expansivity a and heat capacity Cp are discontinuous, at first glance it is not unreasonable to characterize the transformation occurring at Tg as a second-order phase transformation. After all, recall that, by definition, second-order phase transitions require that the properties that depend on the first derivative of the free energy G such as... 

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