Tangents, free-energy densities

Finally, use the free-energy density vs. composition curves and work the tangent-to-curve construction in reverse. Using the result that Aga = —12 x 107Jm-3, the corresponding tangent to the a-phase curve will be at about 33 at. % B. 

A useful tool for stability calculations is the tangent plane distance [20], Let us first define this generically for a system with (a vector of) densities p, free energy density/(p) and chemical potentials p(p) = V/(p) this... 

The whole composition range, 02 = 0 1, is subdivided into five sections by the binodal compositions 0 and 0f anc the spinodal compositions 0 and 0. The binodal points A and B are defined as points of common tangent to the free energy density curve. For compositions between 0 and 0 and between 0 and 1, the free energy of the mixture increases whenever it separates into two phases of... 

Figure 2.13. The excess free energy density rp p, T) as a fimction of the density p for a fluid below its critical point. The double tangent construction defines the equilibrium densities of gas and liquid phases.

Figure 13.3 Dependence of Gibbs free energy density (per atom) on composition for different temperatures Q is the initial composition. The mole fractions C and Cp are linked by the parallel tangents rule they depend on the nuclear radius r and provide the minimization of AG (energy of the system) with respect to concentrations for...

The equihbrium concentrations in the new phase and in the ambient parent phase are determined by the rule of the parallel (not common) tangents drawn from lines of concentration dependencies of Gibbs free energy densities on concentrations for new and old phases [61]. 

Fig. 6. Schematic representation of the surface excess free energy Aa as a function of density. The double tangent construction defines the densities of the equilibrium coexisting phases, pgas and piiquid- The slope of the double tangent construction is equal to the chemical potential m. Ao for a particular density is defined as the difference between the free energy density curve and the double tangent line.

Figure A2.5.9. (Ap), the Helmholtz free energy per unit volume in reduced units, of a van der Waals fluid as a fiinction of the reduced density p for several constant temperaPires above and below the critical temperaPire. As in the previous figures the llill curves (including the tangent two-phase tie-lines) represent stable siPiations, the dashed parts of the smooth curve are metastable extensions, and the dotted curves are unstable regions. See text for details.

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... 

Figure 5. Examples of moment free energy (70) for Flory-Huggins theory of length-polydisperse polymers, with one moment density, p, retained. The parent is of the Schulz form (65), with pf = 0.03, Lu = 100 (hence p = p, /Lv = 3 x 10-4), and a = 2 (hence Lw = 150) the point pt = pj° is marked by the filled circles. In plot (a), the value of x = 0.55 is sufficiently small for the parent to be stable The moment free energy is convex. Plot (b) shows the cloud point, % 0.585, where the parent lies on one endpoint of a double tangent the other endpoint gives the polymer volume fraction p, in the shadow phase. Increasing x further, the parent eventually becomes spinodally unstable [x 0.62, plot (c)]. Note that for better visualization, linear terms have been added to all free energies to make the tangent at the parent coincide with the horizontal axis.

For the case of dumbbells formed from tangent spheres, L = ct, it is also possible to pack the dumbbells in orientationally disordered structures in which the spheres of the dumbbell lie on an fee lattice but the centers of mass form an aperiodic structure. This idea has been explored more extensively for two-dimensional dumbbells by Wojciechowski et al. [241,242]. The configurational degeneracy of the aperiodic structure renders it more stable than the orientationally ordered structures at all densities, even though the effect upon the equation of state and the free energy without the contribution from the degeneracy is quite small [60]. The freezing of hard dumbbells into such structures has also been studied by Bowles and Speedy [243],... 

Figure 10. A portion of one subcritical isotherm. The dashed line shows the double tangent, obtained numerically after local interpolation in the vicinity of the coexisting densities the squares mark the resulting liquid and gas phases. The slope of the double tangent gives the vapor pressure in view of the high precision of the free energies, the vapor pressure is obtained very precisely.

Figure 13.6 Effect of size on equilibrium state and solubility limits found by the common tangent method. Curves Cs,oo(C) and Cl,co(C) characterize the energy density dependence on composition for solid and liquid phases in bulk form, respectively. Cs,r(C) is the Gibbs free energy of the solid...

In a series of papers [7,106,107], we have combined our EoS model with the density gradient approximation of inhomogeneous systans [99-105]. In Refs. [7,106,107], we have addressed in three alternative ways the problon of consistency and equivalence of the various methods of calculating the interfacial tension. In the first case [106], we have simulated the number density profile across the interface with the classical hyperbolic tangent expression [92] (Equation 2.138). In the second case [7], this profile was obtained from the free-energy minimization condition [103,105]. 

Macroscopically, the surface is assumed to be smooth (as indicated by the dashed line in the figure) and to have a local surface energy density Us at any point on the surface, determined by the orientation of the tangent plane and the level of elastic strain in the crystal at that point. A quantitative interpretation of the free energy of a strained crystal surface in terms of the... 

From Helfand s theory, it follows that the free energy of a system will be less with a diffuse interface. As a measure of the interphase thickness, the theory assumes the segment between the points of intersection of the inflectional tangent to the density gradient curve with density levels pi and p2 ... 

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