Gibbs fluid

Figure 8-7. Solids under non-hydrostatic stress aud the surrounding Gibbs-fluids L.

The surrounding fluid (Fig. 8-7) serves two purposes 1) it transmits the pressure to stress-load the surface and 2) it allows the surface to equilibrate chemically and thus provides juL in Eqn. (8.61) with physical meaning. Ideally, the Gibbs fluid has a vanishing buffer capacity for the solid so that after a change in an, the fluid becomes resaturated with respect to the solid before a noticeable amount of the solid or its surface dissolves. The key to verify Gibbs relation for solids under non-hydrostatic stress is therefore the existence of such an idealized fluid. 

Solid electrolytes may have the requisite properties of a Gibbs fluid [W. Durham, H. Schmalzried (1987)] if 1) their conducting ion corresponds to an atomic component of the solid under stress and 2) they exhibit significant mechanical strength. Topical stress energy densities correspond to electrical potentials in the millivolt range. In order to establish them, only a small fraction of a surface monolayer of the electrolyte needs to dissolve during its equilibration with the stressed solid and... 

The Gibbs ensemble method has been outstandingly successfiil in simulating complex fluids and mixtures. 

Panagiotopoulos A Z 1992 Direot determination of fluid phase equilibria by simulation in the Gibbs ensemble a review Mol. SImul. 9 1 -23... 

Panagiotopoulos A Z 1987 Adsorption and oapillary oondensation of fluids in oylindrioal pores by Monte Carlo simulation in the Gibbs ensemble Mol. Phys. 62 701-19... 

Equations 54 and 58 through 60 are equivalent forms of the fundamental property relation apphcable to changes between equihbtium states in any homogeneous fluid system, either open or closed. Equation 58 shows that ff is a function of 5" and P. Similarly, Pi is a function of T and C, and G is a function of T and P The choice of which equation to use in a particular apphcation is dictated by convenience. Elowever, the Gibbs energy, G, is of particular importance because of its unique functional relation to T, P, and the the variables of primary interest in chemical technology. Thus, by equation 60,... 

The definitions of enthalpy, H, Helmholtz free energy. A, and Gibbs free energy, G, also give equivalent forms of the fundamental relation (3) which apply to changes between equiUbrium states in any homogeneous fluid system ... 

The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment or all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formahsm of excess properties is analogous to that of the residual properties. 

Surface Excess With a Gibbs dividing surface placed at the surface of the solid, the surface excess of component i, F (moVm"), is the amount per unit area of solid contained in the region near the surface, above that contained at the fluid-phase concentration far from the surface. This is depicted in two ways in Fig. 16-4. The quantity adsorbed per unit mass of adsorbent is... 

Adsorbed-Solution Theoiy The common thennodynamic approach to multicomponent adsorption treats adsorption equilibrium in a way analogous to fluid-fluid equilibrium. The theory has as its basis the Gibbs adsorption isotherm [Young and Crowell, gen. refs.], which is... 

Another example of phase transitions in two-dimensional systems with purely repulsive interaction is a system of hard discs (of diameter d) with particles of type A and particles of type B in volume V and interaction potential U U ri2) = oo for < 4,51 and zero otherwise, is the distance of two particles, j l, A, B] are their species and = d B = d, AB = d A- A/2). The total number of particles N = N A- Nb and the total volume V is fixed and thus the average density p = p d = Nd /V. Due to the additional repulsion between A and B type particles one can expect a phase separation into an -rich and a 5-rich fluid phase for large values of A > Ac. In a Gibbs ensemble Monte Carlo (GEMC) [192] simulation a system is simulated in two boxes with periodic boundary conditions, particles can be exchanged between the boxes and the volume of both boxes can... 

In this section we review several studies of phase transitions in adsorbed layers. Phase transitions in adsorbed (2D) fluids and in adsorbed layers of molecules are studied with a combination of path integral Monte Carlo, Gibbs ensemble Monte Carlo (GEMC), and finite size scaling techniques. Phase diagrams of fluids with internal quantum states are analyzed. Adsorbed layers of H2 molecules at a full monolayer coverage in the /3 X /3 structure have a higher transition temperature to the disordered phase compared to the system with the heavier D2 molecules this effect is... 

This notation admits of generalisation (Gibbs, 1876). The total energy of a homogeneous fluid is a continuous and single-valued function of the masses mi, m2, m3,. . mh of its constituents, of the total volume Y, and the total entropy S ... 

In his original demonstration Gibbs (1874) showed that the surface layer may be considered as a third phase having specific values of density, energy, and entropy, and further that the results of the theory are quite independent of the actual extent of the capillary layer and the way in which it merges into the free fluids on either side. As a matter of fact, the transition... 

The mapping (7) introduces the unknown interface shape explicitly into the equation set and fixes the boundary shapes. The shape function h(x,t) is viewed as an auxiliary function determined by an added condition at the melt/crystal interface. The Gibbs-Thomson condition is distinguished as this condition. This approach is similar to methods used for liquid/fluid interface problems that include interfacial tension (30) and preserves the inherent accuracy of the finite element approximation to the field equation (27)... 

The Gibbs Ensemble MC simulation methodology [17-19] enables direct simulations of phase equilibria in fluids. A schematic diagram of the technique is shown in Fig. 10.1. Let us consider a macroscopic system with two phases coexisting at equilibrium. Gibbs ensemble simulations are performed in two separate microscopic regions, each within periodic boundary conditions (denoted by the dashed lines in Fig. 10.1). The thermodynamic requirements for phase coexistence are that each... 

In summary, the Gibbs ensemble MC methodology provides a direct and efficient route to the phase coexistence properties of fluids, for calculations of moderate accuracy. The method has become a standard tool for the simulation community, as evidenced by the large number of applications using the method. Histogram reweighting techniques (Chap. 3) have the potential for higher accuracy, especially if... 

Gibbs ensemble. Good for obtaining a few points for subcritical phase coexistence between phases of moderate densities does not provide free energies directly. Primarily used to study fluid (disordered) phases. Is a standalone approach, and requires modest programming and computational effort to set up and equilibrate the multiple simulation boxes. Provides accurate coexistence points at intermediate temperatures below the critical point but with sufficient thermal mobility to equilibrate. 

Kristof, T. Liszi, J., Application of a new Gibbs ensemble Monte Carlo method to site-site interaction model fluids, Mol. Phys. 1997, 90, 1031-1034... 

Kiyohara, K. Spyriouni, T. Gubbins, K. E. Panagiotopoulos, A. Z., Thermodynamic scaling Gibbs ensemble Monte Carlo a new method for determination of phase coexistence properties of fluid, Mol. Phys. 19%, 89, 965-974. 

The structure of hydrogels that do not contain ionic moieties can be analyzed by the Flory Rehner theory (Flory and Rehner 1943a). This combination of thermodynamic and elasticity theories states that a cross-linked polymer gel which is immersed in a fluid and allowed to reach equilibrium with its surroundings is subject only to two opposing forces, the thermodynamic force of mixing and the retractive force of the polymer chains. At equilibrium, these two forces are equal. Equation (1) describes the physical situation in terms of the Gibbs free energy. 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

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