Gibbs-Duhem integration coexistence lines

While the main driving force in [43, 44] was to avoid direct particle transfers, Escobedo and de Pablo [38] designed a pseudo-NPT method to avoid direct volume fluctuations which may be inefficient for polymeric systems, especially on lattices. Escobedo [45] extended the concept for bubble-point and dew-point calculations in a pseudo-Gibbs method and proposed extensions of the Gibbs-Duhem integration techniques for tracing coexistence lines in multicomponent systems [46]. 

The silicon melting line has been determined using a coexistence simulation292 and is able to predict a decrease in Tm with increase in pressure, consistent with experiment. Vega et al.2S9 carry out Gibbs-Duhem integration to determine the coexistence line of water, and the change in the coexistence line... 

Once a state point of coexistence is established, additional state points can be determined expeditiously through application of the Gibbs-Duhem integration method [48,85,86]. In this approach a differential equation for the coexistence line is used to guide the establishment of state points away from the known coexistence point. The most well known such formula is the Clapeyron equation [41]... 

The n= 12 soft sphere model is the high-temperature limit of the 12-6 Lennard-Jones (LJ) potential. Agrawal and Kofke [182] used this limit as the starting point for another Gibbs-Duhem integration, which proceeded to lower temperatures until reaching the solid-liquid-vapor triple point. The complete solid-fluid coexistence line, from infinite temperature to the triple point, can be conveniently represented by the empirical formula [182]... 

Gibbs-Duhem Integration Tracing Coexistence Lines A. Concept Formalism... 

Figure 3. Vapor-liquid coexistence line for the Lennard-Jones model. Solid line is presently the best determination of the phase boundary [50] triangles [86] and circles [59] are Gibbs ensemble data. Dashed line is obtained from Gibbs-Duhem integration beginning with the low-temperature (rightmost) Gibbs ensemble datum. Error bars on the true line are the stability analysis prediction of how the error in the initial datum propagates through the integration series.

There are many different routes for calculating phase equilibria that are covered in detail in other chapters of this volume thermodynamic scaling Monte Carlo (chapter by Valleau), Gibbs-Duhem integration along coexistence lines (chapter by Kofke), and pseudo-ensemble methods (chapter by de Pablo and Escobedo). Thus these methods are not discussed here. 

The Gibbs-Duhem integration method excels in calculations of solid-fluid coexistence [48,49], for which other methods described in this chapter are not applicable. An extension of the method that assumes that the initial free energy difference between the two phases is known in advance, rather than requiring it to be zero, has been proposed by Meijer and El Azhar [51]. The procedure has been used in [51] to determine the coexistence lines of a hard-core Yukawa model for charge-stabilized colloids. 

In Figure 1 b a different construction is described. In this situation two coexistence lines are being traced by the Gibbs-Duhem method, each line having one phase (a) in common. The point where the fi-ct coexistence coincides with the -a. coexistence again forms a triple point. Subsequent integration then follows coexistence between the and p phases. It might be... 

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