Fluid structure, equilibrium theory

Equilibrium Theory of Fluid Structure. In all the theoretical work reported herein, we assume that the particles Interact with pair additive forces whose pair potentials can be approximated by... 

Linear response theory [152] is perfectly suited to the study of fluid structures when weak fields are involved, which turns out to be the case of the elastic scattering experiments alluded to earlier. A mechanism for the relaxation of the field effect on the fluid is just the spontaneous fluctuations in the fluid, which are characterized by the equilibrium (zero field) correlation functions. Apart from the standard technique used to derive the instantaneous response, based on Fermi s golden rule (or on the first Bom approximation) [148], the functional differentiation of the partition function [153, 154] with respect to a continuous (or thermalized) external field is also utilized within this quantum context. In this regard, note that a proper ensemble to carry out functional derivatives is the grand ensemble. All of this allows one to gain deep insight into the equilibrium structures of quantum fluids, as shown in the works by Chandler and Wolynes [25], by Ceperley [28], and by the present author [35, 36]. In doing so, one can bypass the dynamics of the quantum fluid to obtain the static responses in k-space and also make unexpected and powerful connections with classical statistical mechanics [36]. 

The equilibrium thermodynamic and structural properties of coUoidal dispersions may be treated in the same way as in the case of simple liquids by considering the colloidal particles as "supramolecules dispersed in a continuous (but fluctuating) back-ground. The potential which for the case of fluctuating forces replaces the interaction potential between molecules (in vacuo) is the potential of the average forces which act between the dispersed particles. This effective interaction is the input for statistical mechanical theories. Therefore statistical mechanical theories developed for simple fluids can be applied to colloidal dispersions. The theoretical basis for such a treatment was given by Onsager and Me Millan and Mayer. In recent years concepts of liquid state theory have been applied successfully to understand the behavior of concentrated colloidal dispersions. ... 

Equilibrium theories of classical fluids are often formulated as integral equations for the distribution functions, e.g., AA(r) giving the probability distribution of AA pairs at a specified separation in water. The Pratt-Chandler (PC) theory was the first theory of hydrophobic effects largely built on that conventional basis. A less conventional feature was that the PC theory avoided calculating separately the structure of water in the absence of hydrophobic solutes, much in the spirit of the scaled particle models. Instead, the measured goo(r) was used as input to the theoretical calculation of gAA(r). The idea was to use what is known about pure water to make predictions about hydrophobic effects. The ITM makes this heuristic point of view explicit. Moreover, the success of the simplest two-moment ITM provides important support for... 

A great deal of research remains to be done in this area. We are currently extending in the study of spatial correlations in the non-equilibrium fluids to time correlations with the hope of establishing a correspondence between MD and fluctuating hydrodynamic theory. We are also using these systems to study the roles of viscosity and conductivity in fluid behavior under different external constraints. Finally, we plan to continue our research into the formation of spatial structures in fluids. 

Theories of electron mobility are intimately related to the state of the electron in the fluid. The latter not only depends on molecular and liquid structure, it is also circumstantially influenced by temperature, density, pressure, and so forth. Moreover, the electron can simultaneously exist in multiple states of quite different quantum character, between which equilibrium transitions are possible. Therefore, there is no unique theory that will explain electron mobilities in different substances under different conditions. Conversely, given a set of experimental parameters, it is usually possible to construct a theoretical model that will be consistent with known experiments. Rather different physical pictures have thus emerged for high-, intermediate- and low-mobility liquids. In this section, we will first describe some general theoretical concepts. Following that, a detailed discussion will be presented in the subsequent subsections of specific theoretical models that have been found to be useful in low- and intermediate-mobility hydrocarbon liquids. 

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 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... 

William Russel May I follow up on that and sharpen the issue a bit In the complex fluids that we have talked about, three types of nonequilibrium phenomena are important. First, phase transitions may have dynamics on the time scale of the process, as mentioned by Matt Tirrell. Second, a fluid may be at equilibrium at rest but is displaced from equilibrium by flow, which is the origin of non-Newtonian behavior in polymeric and colloidal fluids. And third, the resting state itself may be far from equilibrium, as for a glass or a gel. At present, computer simulations can address all three, but only partially. Statistical mechanical or kinetic theories have something to say about the first two, but the dynamics and the structure and transport properties of the nonequilibrium states remain poorly understood, except for the polymeric fluids. 

The Molecular Layer Structure Theory (MLST) was first developed [4, 5] to study the vapour liquid equilibrium and surface tension of many substances over a wide range of temperature. In this method, the fluid is considered as parallel molecular layers, whose surface densities are different for inhomogeneous fluids. Similar to the DFT method, we define the following grand potential ... 

Leutheusser 1984 Bentzelius et al. 1984), has stirred both excitement and controversy (Mezei et al. 1987 Richter et al. 1991 Retry et al. 1991 Schonhals et al. 1993 Mezei 1991 Kim and Mazenko 1992). While the details of the mode-coupling theory are beyond the scope of this chapter, the main idea is that at high fluid densities there is a nonlinear feedback mechanism by which fluctuations in the structure (or local density) of the fluid become arrested and cannot relax to equilibrium. The point at which this occurs is then a purely dynamic glass transition. 

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