Stability of a phase

This section describes the necessary conditions for an arbitrary system to remain stable in a specified uniform phase under the given environmental conditions. In order for a 

Minimum work required to change the state of a part P in the closed system. 

State to be stable, the system does not change spontaneously, i.e., it can change only if a positive work is applied to it from the environment [1-5]. 

Because the entire system is isolated, we also have the conditions 

Because the entropy of an isolated system must increase (or stay constant) due to the second law of thermodynamics, the inequality 

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As is well recognized, various macroscopic properties such as mechanical properties are controlled by microstructure, and the stability of a phase which consists of each microstructure is essentially the subject of electronic structure calculation and statistical mechanics of atomic configuration. The main subject focused in this article is configurational thermodynamics and kinetics in the atomic level, but we start with a brief review of the stability of microstructure, which also poses the configurational problem in the different hierarchy of scale. 

So far we have considered only the stability of a phase with respect to perturbations at constant U and F. A similar calculation may be made keeping T and V constant. In this case, instead of calculating the perturbation of the entropy we consider the Helmholtz free energy, F, and write the molar free energy / as a function of T and v. The molar volumes are perturbed by b v and h"v as in (15.21) but since the perturbation is isothermal 8 T — b"T = 0. We then obtain a formula... 

The method is accurate, but the result is limited to low surfactant concentrations (the linear part) and solute concentrations at saturation. It should also be noted that the stability of a phase depends on the properties of the precipitating phase as well. For water-surfactant-alcohol systems the precipitating phases may be an alcohol-rich phase (L2) or a lamellar phase (D), and the data indicate that the linear part where Equation 6.9 is fulfilled coincides with the part where the precipitating phase is the alcohol-rich L2 phase. ... 

In general, the stability of a phase-transfer catalyst is a function of cation structure, presence of anions, type of solvent, concentration, and temperature. Degradation of catalysts under PTC conditions may occur. For instance, ammonium and phospho-nium salts may be subject to decomposition by internal displacement (usually at temperatures of 100-200 °C) ... 

Vas2] Calculations, Mdssbauer spectroscopy Stability of a phase... 

Piezochromism or mechanochromism. The stability of a phase is a function of the temperature and pressure of the system. 

Theorem 6.- The stability of a phase requires the following conditions on the thermodynamic coefficients (whatever the ensemble of variables chosen) ... 

It is noted in Sections XVII-10 and 11 that phase transformations may occur, especially in the case of simple gases on uniform surfaces. Such transformations show up in q plots, as illustrated in Fig. XVU-22 for Kr adsorbed on a graphitized carbon black. The two plots are obtained from data just below and just above the limit of stability of a solid phase that is in registry with the graphite lattice [131]. 

A major drawback of MD and MC techniques is that they calculate average properties. The free energy and entropy fiinctions caimot be expressed as simple averages of fimctions of the state point y. They are directly coimected to the logaritlun of the partition fiinction, and our methods do not give us the partition fiinction itself Nonetheless, calculating free energies is important, especially when we wish to detennine the relative thenuodynamic stability of different phases. How can we approach this problem ... 

The nematic to smectic A phase transition has attracted a great deal of theoretical and experimental interest because it is tire simplest example of a phase transition characterized by tire development of translational order [88]. Experiments indicate tliat tire transition can be first order or, more usually, continuous, depending on tire range of stability of tire nematic phase. In addition, tire critical behaviour tliat results from a continuous transition is fascinating and allows a test of predictions of tire renonnalization group tlieory in an accessible experimental system. In fact, this transition is analogous to tire transition from a nonnal conductor to a superconductor [89], but is more readily studied in tire liquid crystal system. 

Frenkel D 1988 Thermodynamic stability of a smectic phase in a system of hard rods Nature 332 822-3... 

A thermodynamically stable system conserves energy. Thus, by monitoring the potential energy one can confirm that a stable (and productive) phase of the simulation has begun. Absence of systematic drift in computed averages is often used as a check on the stability of a Monte Carlo trajectory. Fluctuations in the energy... 

Phenomena at Liquid Interfaces. The area of contact between two phases is called the interface three phases can have only aline of contact, and only a point of mutual contact is possible between four or more phases. Combinations of phases encountered in surfactant systems are L—G, L—L—G, L—S—G, L—S—S—G, L—L, L—L—L, L—S—S, L—L—S—S—G, L—S, L—L—S, and L—L—S—G, where G = gas, L = liquid, and S = solid. An example of an L—L—S—G system is an aqueous surfactant solution containing an emulsified oil, suspended soHd, and entrained air (see Emulsions Foams). This embodies several conditions common to practical surfactant systems. First, because the surface area of a phase iacreases as particle size decreases, the emulsion, suspension, and entrained gas each have large areas of contact with the surfactant solution. Next, because iaterfaces can only exist between two phases, analysis of phenomena ia the L—L—S—G system breaks down iato a series of analyses, ie, surfactant solution to the emulsion, soHd, and gas. It is also apparent that the surfactant must be stabilizing the system by preventing contact between the emulsified oil and dispersed soHd. FiaaHy, the dispersed phases are ia equiUbrium with each other through their common equiUbrium with the surfactant solution. 

Last in this seetion on lattiee ehain models, let us eite the somewhat different approaeh of Jennings et al. [134], who model the amphiphiles as single-site partieles on a lattiee but surround them with long hydrophobie ehains (of ehain length up to iV = 80). Their study foeuses on the influenee of amphiphiles on the eonformations of nonpolar polymers. They report phenomena sueh as amphiphile-indueed polymer eollapse and the stabilization of lamellar phases. 

We have also studied the stability of bicontinuous phases for a different function describing the surfactant, g[0(r)]. We have used the following form of g[0(r)] ... 

In summary, we have demonstrated the possibility of calculating the phase stability of a magnetic random alloy from first principles by means of LMTO-CPA theory. Our calculated phase diagram is in good agreement with experiment and shows a transition from the partially ordered a phase to an hep random alloy at 85% Co concentration. 

Figure 5. Diagram giving the relative stability of the various atomic configurations shown in Fig. 4 as a function of the d band-filling Nj- From the second to the fifth line relative stability of the F and N sites for the monomer, dimer, A trimer and B trimer. On the sixth and seventh lines relative stability of A and B triangles at N and F sites. The relative stability of HCP and FCC bulk phases is given for comparison in the first line.

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