Free-energy density

As for crystals, tire elasticity of smectic and columnar phases is analysed in tenns of displacements of tire lattice witli respect to the undistorted state, described by tire field u(r). This represents tire distortion of tire layers in a smectic phase and, tluis, u(r) is a one-dimensional vector (conventionally defined along z), whereas tire columnar phase is two dimensional, so tliat u(r) is also. The symmetry of a smectic A phase leads to an elastic free energy density of tire fonn [86]... 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

A phase itself, the amplitude of the density modulation is constant and twist and splay distortions are forbidden, thus the expression for the free energy density simplifies to equation (C2.2.10). 

Figure 2. A schematic of the free energy density of an aperiodic lattice as a function of the effective Einstein oscillator force constant a (a is also an inverse square of the locahzation length used as input in the density functional of the liquid). Specifically, the curves shown characterize the system near the dynamical transition at Ta, when a secondary, metastable minimum in F a) begins to appear as the temperature is lowered. Taken from Ref. [47] with permission.

Cohesive energy (or free energy) density parameter (Chaps. XII and XIII). 

This will be the form of the profile if we used a fully local free energy density functional in our calculation. In the case of the HS-B2-approximation the fully local functional would... 

The results of the simple DHH theory outlined here are shown compared with DH results and corresponding Monte Carlo results in Figs. 10-12. Clearly, the major error of the DH theory has been accounted for. The OCP model is greatly idealized but the same hole correction method can be applied to more realistic electrolyte models. In a series of articles the DHH theory has been applied to a one-component plasma composed of charged hard spheres [23], to local correlation correction of the screening of macroions by counterions [24], and to the generation of correlated free energy density functionals for electrolyte solutions [25,26]. The extensive results obtained bear out the hopeful view of the DHH approximation provided by the OCP results shown here. It is noteworthy that in... 

The mean-field expression for the free-energy density of the polymer solution is therefore [ 13,14]... 

In the perfectly ordered crystalline ground state, all polymer bonds are parallel and no solvent-polymer contacts are present. If we ignore disorder (vacancies, kinks) in the polymer crystal at finite temperatures, the free-energy density of the crystalline state is zero. 

In a pure homopolymer system, the free-energy density only depends on Ec (the quantity that determines the chain rigidity) and Ep (the quantity that determines the tendency of backbone chains to form parallel, close-packed structures). Let us first consider the relative stability of the pure polymer melt and the polymer solid in the limit of infinitely long chains. In that case, we... 

We have introduced a gauge fixing term, with the limit e —> 0 being taken after the calculations are carried out.In Eq.(5) A stands for the transverse gauge field and (j) is defined as ip = 4>exp(i y). To obtain the free energy density, Veff = T jV (effective potential), we introduce a shifted field 4> — 4> + and split the Hamiltonian into two parts ... 

In the absence of electromagnetic and gluonic fields the free energy density F of a homogeneous color superconducting phase near Tc is [10, 17]... 

In the absence of magnetic and chromomagnetic fields and in the homogeneous limit the free energy density becomes... 

The first ingredient in any theory for the rheology of a complex fluid is the expression for the stress in terms of the microscopic structure variables. We derive an expression for the stress-tensor here from the principle of virtual work. In the case of flexible polymers the total stress arises to a good approximation from the entropy of the chain paths. At equilibrium the polymer paths are random walks - of maximal entropy. A deformation induces preferred orientation of the steps of the walks, which are therefore no longer random - the entropy has decreased and the free energy density/increased. So... 

Here f(c) is the free energy density of a hypothetical completely homogeneous system with volume fraction c. It is understood that c=c(r). For small fluctuations, one can expand f(c) about c, the bulk composition of the mixture. Keeping only terms to second order and invoking conservation of mass, one may obtain an expression for AF, the difference between the free energy of homogeneous and inhomogeneous mixtures ... 

In considering the selection of anodes for high energy density (HED) storage (or secondary) batteries (SB), we note that there are some 19 metals whose free-energy density (TED) of reaction with oxidants such as O2, Cl2, and F2 are higher than those of Zn with the same oxidants. Most of these metals react violently with water. The remainder are passivated by water. Therefore other electrolytes must be considered for these metals, based on non-aqueous, molten salt, or solid-state ionic conductors. Much experimental work has been carried out during the last two decades on primary and secondary batteries based on anhydrous electrolytes, aimed at utilization of the active metals. 

As a specific example, consider a blend (discussed in Section III.B) of chains of N segments and 2 chains of N2 segments. The free-energy density... 

Perhaps the most important conclusion arising from a study of such models is that the projected free energy density of a uniform vicinal surface with slope s is given by the familiar Gruber-Mullins " expression ... 

The l.h.s. simply evaluates H in this limit and the r.h.s. is the surface area LyNsW of the flat reference plane times the projected free energy density J s) for a uniform system with Slope s = Mw. Thus we find... 

If energetics is dominated by a spatially uniform magnetization M, the spin part of the free energy density in the magnetic field H can be written in the form... 

Having the hole energies, the Helmholtz free energy density FC[M] can be evaluated according to the standard procedure for the Fermi gas. By minimizing F[Af] = Fs[Af] + FC[M] with respect to M at given T, H, and hole concentration p, one obtains M(T, H) as a solution of the mean-field equation,... 

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