First-order thermodynamic transition

The liquid crystal melt, which comes into being at the glass-rubber transition or at the crystal-melt transition, may have several phase states (Mesophases) one or more smectic melt phases, a nematic phase and sometimes a chiral or cholesteric phase the final phase will be the isotropic liquid phase, if no previous decomposition takes place. All mesophase transitions are thermodynamically real first order effects, in contradistinction to the glass-rubber transition. A schematic representation of some characteristic liquid crystal phase structures is shown in Fig. 6.13, where also so-called columnar phases formed from disclike molecules is given. 

Ehrenfest s classification (see [11]) into first-order and second-order transitions is based on thermodynamic criteria. First-order transitions have discontinuities in the first derivatives of the Gibbs energy with respect to temperature (= entropy) and... 

In this work, we review briefly the phenomenology associated to LLPTs based on results obtained from computer simulations of different systems, such as silica, water, and atomic model systems. When possible, results from computer simulations are compared to available experiments. This work is organized as follows. In the next section, we present the phase diagram of polymorphic liquids supported by many computer simulations and experiments. We review the thermodynamics of first-order phase transitions and show how it is observed in computer Simula tions of polymorphic liquids. The relationship between liquid polymorphism and anomalous properties in liquids is also discussed. The next section also includes a description of glass polymorphism, its relation to liquid polymorphism, and a close comparison between experiments and simulations. In Section III, we describe computer simulation models of systems that present liquid polymorphism, with emphasis on the molecular interactions and common properties of these models that are thought to originate LLPTs. A summary and discussion are presented in Section IV. 

We shall first review very briefly the thermodynamics of first-order phase transitions between ordinary bulk phases at equilibrium. We shall then be able to deseribe in similar terms the closely analogous phase transitions in interfaces. Among these is the Cahn transition, our present subject. 

In Secs. 4.3 and 4.4 we discussed the thermodynamics of the crystal -> Uquid transition. This and other famiUar phase equilibria are examples of what are called first-order transitions. There are other less familiar but also well-known... 

Another well-known thermodynamic result, the Clapeyron equation, applies to first-order transitions (subscript 1) ... 

The aim of the present study is precisely to investigate the thermodynamical properties of an interface when the bulk transition is of first order. We will consider the case of a binary alloy on the fee lattice which orders according to the LI2 (CuaAu type) structure. 

The activation parameters from transition state theory are thermodynamic functions of state. To emphasize that, they are sometimes designated A H (or AH%) and A. 3 4 These values are the standard changes in enthalpy or entropy accompanying the transformation of one mole of the reactants, each at a concentration of 1 M, to one mole of the transition state, also at 1 M. A reference state of 1 mole per liter pertains because the rate constants are expressed with concentrations on the molar scale. Were some other unit of concentration used, say the millimolar scale, values of AS would be different for other than a first-order rate constant. 

First-order phase transitions exhibit hysteresis, i.e. the transition takes place some time after the temperature or pressure change giving rise to it. How fast the transformation proceeds also depends on the formation or presence of sites of nucleation. The phase transition can proceed at an extremely slow rate. For this reason many thermodynamically unstable modifications are well known and can be studied in conditions under which they should already have been transformed. 

Although transition across a critical point may proceed without any first-order discontinuity, the fact that there is a change of symmetry implies that the two phases must be described by different functions of the thermodynamic variables, which cannot be continued analytically across the critical point. The order parameter serves to compensate for the reduction in symmetry. Although it is a regular function of temperature it can develop a discontinuous derivative at the critical temperature. Likewise, several measurable... 

Thermodynamic representation of transitions often represents a challenge. First-order phase transitions are more easily handled numerically than second-order transitions. The enthalpy and entropy of first-order phase transitions can be calculated at any temperature using the heat capacity of the two phases and the enthalpy and entropy of transition at the equilibrium transition temperature. Small pre-tran-sitional contributions to the heat capacity, often observed experimentally, are most often not included in the polynomial representations since the contribution to the... 

The Landau theory predicts the symmetry conditions necessary for a transition to be thermodynamically of second order. The order parameter must in this case vary continuously from 0 to 1. The presence of odd-order coefficients in the expansion gives rise to two values of the transitional Gibbs energy that satisfy the equilibrium conditions. This is not consistent with a continuous change in r and thus corresponds to first-order phase transitions. For this reason all odd-order coefficients must be zero. Furthermore, the sign of b must change from positive to negative at the transition temperature. It is customary to express the temperature dependence of b as a linear function of temperature ... 

Figure 7. Countour map of the thermodynamic potential in the dynamical mass (M) - wave number (q) plane. The absolute minimum is denoted by the cross for given density. We have the first order phase transitions in this calculation.

The Greek indices a,j3= II, B,G) count colors, the Latin indices i = u,d,s count flavors. The expansion is presented up to the fourth order in the diquark field operators (related to the gap) assuming the second order phase transition, although at zero temperature the transition might be of the first order, cf. [17], iln is the density of the thermodynamic potential of the normal state. The order parameter squared is D = d s 2 = dn 2 + dG 2 + de 2, dR dc dB for the isoscalar phase (IS), and D = 3 g cfl 2,... 

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