Free energy density model theory

The model used was inspired by the free energy density functioned theory developed by Telo da Geuna and Gub-bins [17] and put into practice by Smit and co-workers [6]. Each particle in the simulation can be identified as one of two species oil or water. Amphiphiles are nothing more... 

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... 

Given the expression for K(T), one can construct an EOS by modeling the excess free energy density by = HS + u + ID + DI + DD + where is summed over contributions from hard-sphere (HS), ion-ion (II), ion-dipole (ID), dipole-ion (DI), and dipole-dipole interactions (DD), respectively. 4>ex also contains the contribution due to the internal partition function of the ion pair, = — p lnK(T). Pairing theories differ in the terms retained in the expression for ex. 

A simplified version of the theory was presented by Cevc and Marsh.12 They started from the Marcelja—Radic phenomenological treatment, assumed that the polarization constitutes the order parameter, and used the Gruen—Marcelja model to explain the various contributions to the free energy density. 

The complete formulation includes Eqs. (31)—(33) the SCF of Eq. (39) an equation-of-state relating the local free-energy density to the volume fraction and boundary conditions at the surface and in bulk. Unfortunately, the discussion of Section II demonstrates that no single solution theory accurately describes the thermodynamic properties for all concentrations and temperatures. Thus, specialization to particular solution models is necessary. 

The simplest model used to explain the temperature dependence of (Ap) is based on the Landau-de Gennes theory of the isotropic phase. Sluckin and Poniewierski added two surface terms to the free energy density [26]... 

To apply this theory to calculate surface tensions one simply uses the free energy density function derived from equation (2.5.8) in conjunction with the square gradient theory of section 2.4. As usual this leads to a smoothly var3dng density profile between the liquid and vapour phase, which may be visualised in terms of the FOV model as a variation in cell size through the stnface, as shown schematically in figure 2.24. 

EoS models can also be used in the frame of the gradient approximation, such as the Cahn-Hilliard theory [100] of inhomogeneous systems, for the description of surface properties. In the frame of this theory, the Helmholtz s free-energy density r in a one-component inhomogeneous system can be expressed as an expansion of density p and its derivatives ... 

Much of what has been done on the theory of the near-critical interface has been within the framework of the van der Waals theory of Chapter 3, so much of our present understanding of the properties of those interfaces comes from that theory or from some suitably modified or extended version of it. As we shall see, an interface thickens as Hs critical point is approached, and the gradients of denaty and composition in the interface are then small. Thus, the view that the interfadal region may be treated as matter in bulk, with a local free-energy density that is that of a hypothetically uniform fluid of composition equal to the local composition, with an additional term arising from the non-uniformity, and that the latter may be approximated by a gradient expansion, typically truncated in second order, is then most likely to be successful and perhaps even quantitatively accurate. In this section we shall see what the simplest theory of that kind— that which comes from treating simple models in mean-field approximation, as in Chapter 5— yields for the structure and tension of an interface near a critical point. 

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

Random interface models for ternary systems share the feature with the Widom model and the one-order-parameter Ginzburg-Landau theory (19) that the density of amphiphiles is not allowed to fluctuate independently, but is entirely determined by the distribution of oil and water. However, in contrast to the Ginzburg-Landau approach, they concentrate on the amphiphilic sheets. Self-assembly of amphiphiles into monolayers of given optimal density is premised, and the free energy of the system is reduced to effective free energies of its internal interfaces. In the same spirit, random interface models for binary systems postulate self-assembly into bilayers and intro-... 

Gas, cells, 464, 477, 511 characteristic equation, 131, 239 constant, 133, 134 density, 133 entropy, 149 equilibrium, 324, 353, 355, 497 free energy, 151 ideal, 135, 139, 145 inert, 326 kinetic theory 515 mixtures, 263, 325 molecular weight, 157 potential, 151 temperature, 140 velocity of sound in, 146 Generalised co-ordinates, 107 Gibbs s adsorption formula, 436 criteria of equilibrium and stability, 93, 101 dissociation formula, 340, 499 Helmholtz equation, 456, 460, 476 Kono-walow rule, 384, 416 model, 240 paradox, 274 phase rule, 169, 388 theorem, 220. Graetz vapour-pressure equation, 191... 

GB-like approximations [41, 71, 119, 161, 187, 189, 230-233] may be derived from eq (1) by using the concept of dielectric energy density, as in the work of Bucher and Porter [130], Ehrenson [131], and Schaefer and Froemmel [234], As the GB methodology has been extensively reviewed in the recent past [81, 83, 213], we confine our presentation to a very brief discussion of the key aspects of the theory. The polarization free energy in the GB model is defined as... 

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