Coexistence of two phases

On the left-hand side of Fig. 4 we have the normal phase diagram of the binary BE-H2O system. The addition of DEC shifts the two phase equilibria to higher BE concentrations and to lower temperatures. If the addition of a third component is continued beyond the cloud point, eventually three distinct phases appear. Unfortunately the cloud point technique gives us the initial concentration or temperature where unmixing begins but is not suitable to distinguish between the coexistence of two phases and three phases. Also the three phase region depends quite critically on temperature. 

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. 

We attempt below to put the results in the context of a phase separation [4]. The decomposition of l/63l/ i(T, x) into two terms, as it will be discussed below in more details, manifests itself in a broad temperature interval above Tc. It is limited from above by a T that depends on the concentration, x. We consider T defined in this way as a temperature of a 1st order phase transition, which, however, cannot complete itself in spatial coexistence of two phases because of the electroneutrality condition [5]. It was already argued in [4] that such a frustrated 1st order phase transition may actually bear a dynamical character. The fact that a single resonant frequency for the 63Cu nuclear spin is observed in the NMR experiments, confirms this suggestion. Although in what follows, we use the notions of the lattice model [4, 5], even purely electronic models [6-9] for cuprates may reveal a tendency to phase separation. 

Wc have so far studied only perfect gases and have not taken up imperfect gases, liquids, and solids. Before we treat them, it is really necessary to understand what happens when two or more phases are in equilibrium with each other, and the familiar phenomena of melting, boiling, and the critical point and the continuity of the liquid and gaseous states. We shall now proceed to find the thermodynamic condition for the coexistence of two phases and shall apply it to a general discussion of the forms of the various thermodynamic functions for matter in all three states. 

The characteristic feature of all these processes is the coexistence of two phases. According to the phase rule, a two-phase system consisting of a single species is univariant, and its intensive state is determined by the specification of just one intensive property. Thus the latent heat accompanying a phase change is a function of temperature only, and is related to other system properties by an exact thermodynamic equation ... 

Two-dimensional phase diagrams are often displayed in the form of In [p] against 1/7 (at a constant specific amount adsorbed), which provides a convenient way of indicating the conditions for the coexistence of two phases (see Figure 4.3). Indeed, the application of the Phase Rule indicates that when two adsorbed phases coexist in equilibrium, the system has one degree of freedom therefore, at constant... 

In addition to the identification of crystal moditications, SSNMR has been used to monitor reactivity and phase changes in different polymorphic forms. For instance, Harris and Thomas (1991) followed the photochemical conversion of formyl-fran -cinnamic acid with SSNMR (see also Section 6.4). Variable temperature techniques have been used to study the interconversion of four polymorphic modifications of sulphanilamide (/ -amino-benzenesulphonamide), including interpretation of at least some of the molecular motions during the course of the transformation (Frydman et al. 1990). A similar combination was augmented with colourimetric techniques to study the coexistence of two phases in the course of a phase transition (Schmidt et al. 1999). Of course, differences between unsolvated and solvated or hydrated crystal moditications may also be readily characterized by the SSNMR technique, as was done with the anhydrous and monohydrate of oxyphenbutazone (Stoltz et al. 1991). Due to the availability of the crystal structures for both modifications the SSNMR results could be interpreted directly in terms of the different atomic environments, especially for the differences in hydrogen bonding in the presence... 

Fig. 20 is a graphical representation of the above relationships (T as abscissae and p as ordinates). The regions in which a single phase (two degrees of freedom) is stable are,the three portions into which the p, T plane is divided. Each of the curves separating any two of these regions corresponds to the coexistence of two phases (liquid-gas, solid-gas, and solid-liquid, each with 1 degree of freedom). 

One way to see that a transition is discontinuous is to detect a coexistence of two phases, in this case the orientationally ordered and disordered phases, in a temperature interval. This is revealed by time variation of the potential energy of the cluster. In the temperamre region of phase coexistence, each cluster dynamically transforms between the phases, and its potential energy fluctuates around two different mean values (Fig. 4). In an ensemble of clusters, the coexistence of different phases is observable insofar as a fraction of the clusters (e.g., in a beam [17]) can exhibit the structure of one phase, while another fraction takes on the stmcture of another phase. 

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. 

The evidenee for a first-order transition at the second plateau has been more ambiguous. As seen in Fig. 3, the isotherm is not horizontal, and this has sometimes been interpreted as an indication that the transition is second-order rather than first-order. On the other hand, fluctuations have been observed in the surface potential, which is consistent with the coexistence of two phases. 

Diffraction from the monolayer was observed only in a region where the isotherm is steep, so the failure to observe a constant lattice spacing during the main transition can be attributed to the compressibility of the gel phase. None of the measurements shows coexistence of two phases at the gel-solid transition. A sharp jump in correlation length and intensity is consistent with a first-order transition, but there is no abrupt change in lattice parameter and no evidence of hysteresis. 

The equilibrium state of coexistence of two phases is governed by the following set of equations... 

The equilibrium degree of swelling or the gel volume of shows a large change in response to external conditions. Only when the volume change is discontinuously and the coexistence of two phases is experimentally proofed, we should use the term phase transition . By applying the stability criteria on the Flory-Rehner equation one can calculate the conditions of phase transition (binodal and spinodal curve, critical point). For details see e.g. (Onuki 1993). 

Direct chemical synthesis of polymers to probe these mechanisms has shown that there are cases where the. shift of optical absorption may be interpreted by the coexistence of two phases, while also cases exist of a continuous change in the optical absorption, with no isobestic point evident. This can be correlated to the local substitution pattern, where some of the patterns, giving a two-phase behaviour, show a co-operative aspect of the transition. 

The argument regarding the link between local minima in the free energy and the coexistence of two phases over a band of temperarure and pressure (or of any two state variables) is not restricted to only two phases. Just as more than two chemical isomers may coexist, more than two phases of a cluster may coexist. There is nothing essential that prohibits the appearance of multiple minima in the free energy. This is perhaps clearest if the order parameter is not a simple scalar but a vector characterizing more than one property of the system. While a variety of kinds of clusters have been objects of study for surface melting, d5i-i53,i64-i83 example is... 

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