Coexistence conditions

The alternative to direct simulation of two-phase coexistence is the calculation of free energies or chemical potentials together with solution of the themiodynamic coexistence conditions. Thus, we must solve (say) pj (P) = PjjCT ) at constant T. A reasonable approach [173. 174. 175 and 176] is to conduct constant-AT J simulations, measure p by test-particle insertion, and also to note that the simulations give the derivative 3p/3 7 =(F)/A directly. Thus, conducting... 

The simulation of a first-order phase transition, especially one where the two phases have a significant difference in molecular area, can be difficult in the context of a molecular dynamics simulation some of the works already described are examples of this problem. In a molecular dynamics simulation it can be hard to see coexistence of phases, especially when the molecules are fairly complicated so that a relatively small system size is necessary. One approach to this problem, described by Siepmann et al. [369] to model the LE-G transition, is to perform Monte Carlo simulations in the Gibbs ensemble. In this approach, the two phases are simulated in two separate but coupled boxes. One of the possible MC moves is to move a molecule from one box to the other in this manner two coexisting phases may be simulated without an interface. Siepmann et al. used the chain and interface potentials described in the Karaborni et al. works [362-365] for a 15-carbon carboxylic acid (i.e. pen-tadecanoic acid) on water. They found reasonable coexistence conditions from their simulations, implying, among other things, the existence of a stable LE state in the Karaborni model, though the LE phase is substantially denser than that seen experimentally. The re-... 

FIG. 24 Monolayer G-LE coexistence conditions from the simulations of Siepmann et al. (Ref. 369) on a pentadecanoic acid model using Gibbs ensemble Monte Carlo simulation. The filled circles are the simulation results. Experimental results are also shown from Ref. 370 (triangles), Ref. 14 (squares), and Ref. 15 (diamonds). (Reproduced with permission from Ref. 369. Copyright 1994 American Chemical Society.)... 

ADHD is rarely encountered without comorbid conditions and often is underdiagnosed. Between 40% and 75% of patients with ADHD will have one or more comorbidities (e.g., learning disabilities, oppositional defiant conduct, anxiety, or depressive disorders).10 It is important to identify other coexisting conditions in patients with ADHD to assist in initial and ongoing selection of treatment. 

Selection of drug therapy should follow the JNC 7 guidelines, but the treatment approach in some patient populations may be slightly different. In these situations, alternative agents may have unique properties that benefit a coexisting condition, but the data may not be based on evidence from outcome studies in hypertension. 

Considerations in the selection of ARV treatment regimens at both the programme level and at the level of an individual patient should include the potency, side effect profile, the potential for maintenance of future treatment options, the anticipated adherence of the patient population with a regimen, coexistent conditions (e.g., co-infections, metabolic abnormalities), pregnancy or the risk thereof, the use of concomitant medications (i.e. potential drug interactions), the potential for primary acquisition of resistant viral strains, and cost and access (Table 8). 

We know that if pua < pip, then only phase a is stable, whereas if pba > pip, then only phase /3 is stable. The desired coexistence condition is thereby determined from... 

Figure 7.2 Phase coexistence conditions (circles), showing phase of lowest chemical potential as a function of (a) T, (b) P. The pgsis (heavy dots), p qilid (dashes), and pso id (solid line) curves are plotted with slopes (7.21) [or (7.23)] consistent with the expected order (7.22) [or (7.24)] for the derivative slope (dp/dT)P [or (dp/dP)T].

It is evident from these considerations that the coexistence condition (7.20) can only remain satisfied under variations dT or dP if these variations are chosen to insure that... 

As described in Section 7.2.2, the phase-coexistence conditions are expressed in terms of the required matching variations of chemical potential in the two phases. Geometrically, this means that the Giiq(x) and Gsol(x) curves must have a common tangent line, which contacts the Guq curve at xliq and the Gsoi curve at xsoi. Figure 7.15 illustrates this... 

The Clapeyron vector (evidently closely related to the coexistence coordinate a) at saturation cf. Section 11.5) is merely the projection of the GD vector (11.141b) onto the nonsingular space spanned by T), P). In terms of this vector, the coexistence condition (11.154) can be written as the Clapeyron matching condition ... 

The coexistence conditions (12.68) that relate each excess intensive vector RK) to the chosen axis intensities R ) can also be written in terms of the conjugate extensive vectors R ) = X/). With the usual metric relationship between intensive and extensive vectors,... 

To define the tangent plane distance (TPD), note first that by virtue of the coexistence conditions, all phases pW jie on a tangent plane to the free energy surface. Points (p,/) on this tangent plane obey the equation / — p p + II = 0, with p and II the chemical potentials and pressure common to all phases. For a generic phase with density distribution p and free energy/(p), the same expression will have a nonzero value that measures how much below or above the tangent plane it lies. This defines the TPD... 

Substance use disorders (SUDs) are collectively the most common coexisting condition that therapists will see in their patients who present with relationship problems, depression, or anxiety disorders. SUDs are common and often unrecognized, at least in the initial evaluation of the patient and family. This chapter does not propose to be a comprehensive treatise on the management of SUDs. It focuses on some features germane to the collaboration of physicians and nonphysician therapists in treating SUDs. We discuss alcoholism first, as the most common and the prototype of the SUDs, and then touch on selected features of several other types of SUDs. 

The Australian Heart Foundation provides the following guidelines for physicians in determining potentially unfavorable effects on coexisting conditions ... 

Coexisting Condition Asthma Low heart rate (bradycardia)... 

A difficulty with the above scheme is that measurements carried out with various actual gases that approach ideal behavior will lead to slightly dilferent results. A better absolute standard is provided by the so-called triple point of water. As we shall see later, the coexistence conditions of water in the solid, liquid, and vapor state can occur only under a set of precisely controlled, invariant conditions determined by the physical characteristics of H2O. These conditions are completely reproducible all over the world. For consistency with the above absolute temperature scheme the triple point of water is assigned a temperature T (triple point of H2O) = 273.16 K = 7). Then any other absolute temperature is determined through the proportionality T = (P/Pt) 273.16, where P is the pressure at T and Pt is the pressure measured for He in equilibrium with water at its triple point. 

The CH4 s-H hydrate was generated under four-phase coexisting condition by the same procedure as the previous study. In order to determine the four-phase equilibrium pressure precisely, the s-H hydrate was formed or dissociated by the pressure control reported in the previous studyWhen the pressure change became within 0.01 MPa, the system was regarded as the phase equilibrium. Usually, it took about two or three days to establish the equilibrium in the present study. In the high temperature range, it took more than four days to complete the equilibrium. 

In Sect. 2 the coexistence conditions of high polymer mixtures are described. Here we focus on the internal interface between two coexisting phases with a bilayer morphology. The properties of this interface determine phase coexistence characteristics necessary to describe segregation phenomena discussed in Sects. 3 and 4. 

The signs in Eq. (11) are governed by the same convention as for Eq. (8b). It has been also shown [49] that in the neighborhood of Tc the interfacial width w is related to the correlation length calculated at coexistence conditions w= 2 b=2 (( >1)=2 (( >2). In practice the hyperbolic tangent function turns out to be also a very good approximate form in the case of NA NB and -dependent y. 

A new approach, directly determining the coexistence conditions, has been proposed by us [74] and independently by Bruder and Brenn [75], providing an... 

In practical terms the coexistence conditions are determined as follows. A layer of some hundreds of nm thick of one of pure components (B say) is spin cast on a silicon wafer with a native oxide or covered with an evaporated metal. A similar layer of the other component (A) is laid on the top of the precast film B using standard [74] - or modified [91] (for hydrophobic polymers) - floating techniques. It is possible to ensure that the surface segregation and wetting effects do not perturb final phase configuration by arranging the surface preferred component to be located near this surface. Also the substrate may be modified (by metal evaporation) to cancel the possible polymer-substrate interactions... 

Since the interface relaxation method was established it has been used to determine the coexistence conditions for over 20 polymer mixtures [74,75,88,91, 92,95-99]. 

Surface induced spinodal decomposition leads, for properly controlled surface fields, to a two layer structure characteristic for coexisting phases. Hence it may be used to determine the coexisting conditions in a more convenient way that with the interfacial relaxation method as the initial bilayer geometry may be avoided. In practical terms the overall composition of the whole thin film may be much better controlled in experiments involving spinodal decomposition. Therefore in experiments studying the equilibrium composition vs depth pro-... 

Both the equilibrium interfacial width and the coexistence compositions vary with the interaction parameter % (see Eqs. 8 and 10). The parameter % is, in turn, temperature dependent. The coexistence conditions, such as those of Figs. 7 and 8, may be described with different level of precision by various approaches yielding different forms of the % parameter [9]. 

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