State equations. Phase transitions

State equations. Phase transitions Landau (1935, 1937ab Landau and Lifshitz, 1964) postulated that any thermodynamic potential per unit volume can be expanded into a series with respect to the order parameter Q in the vicinity of the critical point (Tc)  

In most cases of structural pheise transitions, there exists a more symmetrical (high-temperature) phase when T Tc. The appearance of the order parameter Q when T Tc is associated with the appearance of a less symmetrical ph2tse. The high-temperature phase is often identified with the disordered one, and the low-temperature phase with the ordered one. 

Let Q be a scalar quantity. More generally, it is a tensor (as for liquid crystals). 

Further, we neglect the temperature dependence of b, as it is much weaker than that of Q, and, what is more, b is multiplied by Q to the 4th power. Condition 2 gives us the equilibrium value Qo  

Substitution of Equation 3 into P quation 8 yields C, = 0-i f when T Tc, 

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State equations. Phase transitions Tlie condilions of phase stability with respect to the order parameter are... 

Lee T D and Yang C N 1952 Statistical theory of equations of state and phase transitions II. Lattice gas and Ising models Phys. Rev. 87 410... 

The inclusion of both three and four-particle correlations in nuclear matter allows not only to describe the abundances oft, h, a but also their influence on the equation of state and phase transitions. In contrast to the mean-field treatment of the superfluid phase, also higher-order correlations will arise in the quantum condensate. 

EDXD spectrum, 307/ electronic structure, 306 equation of state, 308/ phase transition, 306 lanthanum... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

The CS pressures are close to the machine calculations in the fluid phase, and are bracketed by the pressures from the virial and compressibility equations using the PY approximation. Computer simulations show a fluid-solid phase transition tiiat is not reproduced by any of these equations of state. The theory has been extended to mixtures of hard spheres with additive diameters by Lebowitz [35], Lebowitz and Rowlinson [35], and Baxter [36]. 

In the microcanonical ensemble, the signature of a first-order phase transition is the appearance of a van der Waals loop m the equation of state, now written as T(E) or P( ). The P( ) curve switches over from one... 

Although there have been few data collected, postshock temperatures are very sensitive to the models which specify y and its volume dependence, in the case of the Gruneisen equation of state (Boslough, 1988 Raikes and Ahrens, 1979a Raikes and Ahrens, 1979b). In contrast, the absolute values of shock temperatures are sensitive to the phase transition energy Ejp of Eq. (4.55), whereas the slope of the versus pressure curve is sensitive to the specific heat (see, e.g.. Fig. 4.28). 

Besides shear-induced phase transitions, Uquid-gas equilibria in confined phases have been extensively studied in recent years, both experimentally [149-155] and theoretically [156-163]. For example, using a volumetric technique, Thommes et al. [149,150] have measured the excess coverage T of SF in controlled pore glasses (CPG) as a function of T along subcritical isochoric paths in bulk SF. The experimental apparatus, fully described in Ref. 149, consists of a reference cell filled with pure SF and a sorption cell containing the adsorbent in thermodynamic equilibrium with bulk SF gas at a given initial temperature T,- of the fluid in both cells. The pressure P in the reference cell and the pressure difference AP between sorption and reference cell are measured. The density of (pure) SF at T, is calculated from P via an equation of state. 

Nevertheless, previous developments and some of our results prove that the structural properties of several systems with short-range repulsive forces are straightforwardly and sufficiently accurately given by ROZ integral equations. Thermodynamic properties are much more difficult to describe. Reliable tools exist to obtain thermodynamics at high temperatures or for states far from phase transitions. Of particular importance, and far from being solved, are the issues related to phase transitions in partly quenched systems, even for simple models with attractive interactions. It seems that the results obtained by Kierlik et al. [27], may serve as a helpful reference in this direction. 

It has been shown that ab initio total energy DFT approach is a suitable tool for studies of phase equilibria at low temperatures and high pressures even when small energy differences of the order of 0.01 eV/mol are involved. The constant pressure optimization algorithm that has been developed here allows for the calculation of the equation of state for complex structures and for the study of precursor effects related to phase transitions. 

Ruckenstein and Li proposed a relatively simple surface pressure-area equation of state for phospholipid monolayers at a water-oil interface [39]. The equation accounted for the clustering of the surfactant molecules, and led to second-order phase transitions. The monolayer was described as a 2D regular solution with three components singly dispersed phospholipid molecules, clusters of these molecules, and sites occupied by water and oil molecules. The effect of clusterng on the theoretical surface pressure-area isotherm was found to be crucial for the prediction of phase transitions. The model calculations fitted surprisingly well to the data of Taylor et al. [19] in the whole range of surface areas and the temperatures (Fig. 3). The number of molecules in a cluster was taken to be 150 due to an excellent agreement with an isotherm of DSPC when this... 

The study of how fluids interact with porous solids is itself an important area of research [6], The introduction of wall forces and the competition between fluid-fluid and fluid-wall forces, leads to interesting surface-driven phase changes, and the departure of the physical behavior of a fluid from the normal equation of state is often profound [6-9]. Studies of gas-liquid phase equilibria in restricted geometries provide information on finite-size effects and surface forces, as well as the thermodynamic behavior of constrained fluids (i.e., shifts in phase coexistence curves). Furthermore, improved understanding of changes in phase transitions and associated critical points in confined systems allow for material science studies of pore structure variables, such as pore size, surface area/chemistry and connectivity [6, 23-25],... 

Equations of state relate the phase properties to one another and are an essential part of the full, quantitative description of phase transition phenomena. They are expressions that find their ultimate justification in experimental validation rather than in mathematical rigor. Multiparameter equations of state continue to be developed with parameters tuned for particular applications. This type of applied research has been essential to effective design of many reaction and separation processes. 

The best-known examples of phase transition are the liquid-vapour transition (evaporation), the solid-liquid transition (melting) and the solid-vapour transition (sublimation). The relationships between the phases, expressed as a function of P, V and T consitute an equation of state that may be represented graphically in the form of a phase diagram. An idealized example, shown in figure 1, is based on the phase relationships of argon [126]. 

The equation of state (EOS), the composition and the possible occurrence of phase transitions in nuclear matter are widely discussed topics not only in nuclear theory, but are also of great interest in astrophysics and cosmology. Experiments on heavy ion collisions, performed over the last decades, gave new insight into the behavior of nuclear systems in a broad range of densities and temperatures. The observed cluster abundances, their spectral distribution... 

From the above estimations we conclude that is it at least a good approximation to consider only homogeneous phases to describe the quark matter phase. In Fig. 4 we display the pressure as a function of fi for neutral homogeneous quark matter phases. We see that at small // the 2SC phase (dashed line) is favored whereas at large // we find a CFL phase (solid line). Normal quark matter (dotted line) turns out to be never favored. This will be our input for the description of the quark matter phase. Of course, in order to construct a compact star, we also have to take into account the possibility of a hadronic component in the equation of state (EOS). To this end, we take a given hadronic EOS and construct a phase transition to quark matter from the requirement of maximal pressure. This is shown in the left panel of Fig. 5 for an example hadronic EOS [53], At the transition point to the quark-matter phase we directly enter the CFL phase and normal or 2SC quark matter is completely irrelevant in this... 

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