Free energy per atom

Here, n°A is the free energy per atom in a vacancy-free crystal composed of only A-atoms with a flat (zero curvature) surface, G(, = Hy — TSy (vib) is the free energy [exclusive of that due to the mixing entropy, Sy (vib) is the vibrational entropy] to form a vacancy, and the last term is the free energy of mixing due to the entropy... 

Burton (39) has calculated properties of Ar clusters containing up to 87 atoms. He finds that the vibrational entropy per atom becomes constant for about 25 atoms. The entropy per atom for spherical face-centered cubic structures exceeds that of an infinite crystal and reaches a maximum between 19 and 43 atoms. An expression for the free energy of the cluster as a function of size was derived and shows that the excess free energy per atom increases with cluster size up to the largest clusters calculated. Although this approach yields valuable thermodynamic information on small clusters, it does not give electronic information. 

With regard to surface free energy and surface tension, one can simply try to fit the coefficients of an expansion in powers of of the Helmholtz free energy per atom, F N)/N, for several values of N ... 

Among nonpolar residues, the larger amino acids, and in particular the aromatic residues, are overrepresented in the hydrophobic cores of transient PPI surfaces [21, 23, 31-33]. While still lipophilic, aromatic side chains have lower transfer free energies per atom than aliphatic side chains [34—36], Methionine, while not a major component of interfaces, is also often more frequently found there than in permanently exposed or buried surfaces, and also has a low transfer free energy. Methionine is known to interact very favorably with aromatic groups, probably due to the polarizability of both aromatic systems and sulfur atoms. Together, these residues also feature the most flexibility and size among nonpolar residues, and thereby have... 

Now it is also understandable that the exchange of kink atoms with the ambient phase can be used for the definition of the equilibrium, because it requires the same Gibbs free energy per atom as the average disintegration free energy of the crystal per atom, i.e., just equal to the chemical potential of the crystalline phase. 

Figure 4.2 Nucleation at polymorphic mode and frozen-in concentration profile (a) dependence of the Gibbs free energy per atom on the compositions of the old and new phases and (b) frozen-in profile in the diffusion couple, approximately linear in the nucleation region.

In addition, expansions into Taylor series are used both for concentrations and for Gibbs free energies per atom resulting in... 

Here is a thermodynamic factor and g is the Gibbs free energy per atom. 

Here Ag is the compound formation Gibbs free energy per atom (it has been measured for many compounds). As a result, Equation 7.89 can be reduced to the following form ... 

Figure 13.9 g(c) Gibbs free energy (per atom) as a function of composition for old and new phases Ago is the isothermal Gibbs free energy density of the cr-alloy from pure solid components. Q is the initial composition of the parent phase, Cp is the... 

The NFE theory describes a simple metal as a collection of ions that are weakly coupled through the electron gas. The potential energy is volume-dependent but is independent of the position of the electrons. This is valid for both solids and dense liquids. At densities well above that of the MNM transition, we can use effective pair potentials and find the thermophysical properties of metallic liquids with the thermodynamic variational methods usually employed in theoretical treatments of normal insulating liquids. One approach is a variational method based on hard sphere reference systems (Shimoji, 1977 Ashcroft and Stroud, 1978). The electron system is assumed to be a nearly-free-electron gas in which electrons interact weakly with the ions via a suitable pseudopotential. It is also assumed that the Helmholtz free energy per atom can be expressed in terms of the following contributions ... 

The chemical potential for surface evolution is often expressed as the free energy per atom or molecule added to the surface. With that definition, the chemical potential is xfl, where is the atomic or molecular volume of the material being deposited. The chemical potential is also sometimes expressed as the free energy change per mole of material added. Again, the difference introduced in this way is only a constant multiple factor of the expression in (8.8), provided that x is defined with respect to the reference configuration of the material. 

Classical theory of nucleation predicts there is a critical radius, or equivalently a critical number i of atoms, below which precipitates are unstable and will re-disolve into the solid solution and above which precipitates will grow, i is obtained by considering the competition between the interface free energy a and the nucleation free energy per atom AG",... 

The nucleation free energy per atom entering equation 24 is given by [44,45]... 

The free energy per atom as a fimction of temperature is shown in Figure 12.3 for different excess energies. Taking the second derivatives of Equation 12.9... 

We note that since we are considering the Gibbs free energy per atom, per molecule, or per reaction stoichiometry, this is by definition the chemical potential. We shall, however, genaally refer to this quantity as the Gibbs free energy, in line with keeping the nomenclature for all other concepts, which we express in per atomic-scale entities. 

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