Second-order fluid theory

The course of this curve suggests that the theoretical slope two, which should hold for a second order fluid, will be reached only at shear rates or angular frequencies considerably lower than the ones used in the experiments. It thus appears that the interrelations given by the above mentioned equations hold even outside the literal range of validity of the discussed theory. Similar results were obtained for two samples of linear polyethylene. 

Bueche-Ferry theory describes a very special second order fluid, the above statement means that a validity of this theory can only be expected at shear rates much lower than those, at which the measurements shown in Fig. 4.6 were possible. In fact, the course of the given experimental curves at low shear rates and frequencies is not known precisely enough. It is imaginable that the initial slope of these curves is, at extremely low shear rates or frequencies, still a factor two higher than the one estimated from the present measurements. This would be sufficient to explain the shift factor of Fig. 4.5, where has been calculated with the aid of the measured non-Newtonian viscosity of the melt. A similar argumentation may perhaps be valid with respect to the "too low /efi-values of the high molecular weight polystyrenes (Fig. 4.4). 

The results of this second-order perturbation theory for a fluid whose pair potential is the Lennard-Jones 6 12 potential,... 

Figure 2. Equation of state of the 6 12 fluid. The points and curves give computer simulation and the second-order perturbation theory results for seven isotherms that are labelled with the appropriate values of kT/e. For the Lennard-Jones fluid the triple-point reduced temperature ana density are about 0.7 and 0.85, respectively, and the critical-point reduced temperature and density are about 1.30 and 0.30, respectively.

It has been established [99] that the primary normal stress difference exhibits a strong dependence on molecular-weight distribution as predicted from the theory of second-order fluids. Thus, the following expression is known [99] to hold... 

Monomer contribution As an alternative to the mentioned TPTl versions, we will attempt to describe the thermodynamics and structme of the binary reference mixture of monomers by means of a second order perturbation theory, based on an analytical solution of the Mean Spherical Approximation (MSA) of simple fluids [299,317]. In this theory, the free energy of the mixture of monomers is described perturbatively in terms of the properties of an auxiliary fluid which contains only repulsive interactions. We therefore split the full Lennard—clones potential, Vy(r) into repulsive and perturbative contributions as suggested by Barker and Henderson [300], so that the repulsive potential, contains all of the positive part of the Lennard—Jones potential, while the perturbation, wf contains all of the negative region ... 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

A set of equations (15)-(17) represents the background of the so-called second-order or pair theory. If these equations are supplemented by an approximate relation between direct and pair correlation functions the problem becomes complete. Its numerical solution provides not only the density profile but also the pair correlation functions for a nonuniform fluid [55-58]. In the majority of previous studies of inhomogeneous simple fluids, the inhomogeneous Percus-Yevick approximation (PY2) has been used. It reads... 

Second-Order Integral Equations for Associating Fluids As mentioned above in Sec. II A, the second-order theory consists of simultaneous evaluation of the one-particle (density profile) and two-particle distribution functions. Consequently, the theory yields a much more detailed description of the interfacial phenomena. In the case of confined simple fluids, the PY2 and HNC2 approaches are able to describe surface phase transitions, such as wetting and layering transitions, in particular see, e.g.. Ref. 84. 

The main conclusion which can be drawn from the results presented above is that dimerization of particles in a Lennard-Jones fluid leads to a stronger depletion of the proflles close to the wall, compared to a nonassociating fluid. On the basis of the calculations performed so far, it is difficult to conclude whether the second-order theory provides a correct description of the drying transition. An unequivocal solution of this problem would require massive calculations, including computer simulations. Also, it would be necessary to obtain an accurate equation of state for the bulk fluid. These problems are the subject of our studies at present. 

Sec. 4 is concerned with the development of the theory of inhomogeneous partly quenched systems. The theory involves the inhomogeneous, or second-order, replica OZ equations and the Born-Green-Yvon equation for the density profile of adsorbed fluid in disordered media. Some computer simulation results are also given. 

We also have studied fluid distribution in the pore H = 6 (Fig. 12(b)) at Ppq = 4.8147 and at two values of Pp, namely at 3.1136 (p cr = 0.4) and at 7.0026 (pqOq = 0.7 Fig. 12(b)). In this pore, we observe layering of the adsorbed fluid at high values of the chemical potential Pp. The maxima of the density profile pi(z) occur at distances that correspond to the diameter of fluid particles. With an increase of the fluid chemical potential, pore filhng takes place primarily at pore walls, but second-order maxima on the density profile pi (z) are also observed. The theory reproduces the computer simulation results quite well. 

Fig. 27. Phase diagram of an adsorbed film in- the simple cubic lattice from mean-fleld calculations (full curves - flrst-order transitions, broken curves -second-order transitions) and from a Monte Carlo calculation (dash-dotted curve - only the transition of the first layer is shown). Phases shown are the lattice gas (G), the ordered (2x1) phase in the first layer, lattice fluid in the first layer F(l) and in the bulk F(a>). For the sake of clarity, layering transitions in layers higher than the second layer (which nearly coincide with the layering of the second layer and merge at 7 (2), are not shown. The chemical potential at gas-liquid coexistence is denoted as ttg, and 7 / is the mean-field bulk critical temperature. While the layering transition of the second layer ends in a critical point Tj(2), mean-field theory predicts two tricritical points 7 (1), 7 (1) in the first layer. Parameters of this calculation are R = —0.75, e = 2.5p, 112 = Mi/ = d/2, D = 20, and L varied from 6 to 24. (From Wagner and Binder .)...

Henderson D, Sokolowski S, Wasan DT (1997) Second order Percus-Yevick theory for a confined hard sphere fluid. J Stat Phys 89 233-247... 

We refer to Eq. (92) as a second-order or a pair theory. It is a pair theory in the sense that two fluid particles are considered. The HAB scheme is a singlet theory in the sense that only one fluid particle is considered. 

Using the theory developed by Chapman-Enskog (see Ref. 14), a hierarchy of continuum fluid mechanics formulations may be derived from the Boltzmann equation as perturbations to the Maxwellian velocity distribution function. The first three equation sets are well known (1) the Euler equations, in which the velocity distribution is exactly the Maxwellian form (2) the Navier-Stokes equations, which represent a small deviation from Maxwellian and rely on linear expressions for viscosity and thermal conductivity and (3) the Burnett equations, which include second order derivatives for viscosity and thermal conductivity. 

The Cahn approach describing simple fluid mixtures has been adopted by a mean field theory developed for polymer mixtures by Nakanishi and Pincus [61] and Schmidt and Binder [15] and is presented in the next section. The mean field theory and its various extensions [7] have been successfully used to describe much of the experimental segregation isotherm z (())<, ) data obtained so far [16, 92,120,145,165-167,169-175]. It allows us not only to distinguish isotherms z (J characteristic for partial and complete (first and second order) wetting but also to determine surface free energy parameters useful in predicting surface phase diagrams for the studied mixtures. 

Feigenbaum went on to develop a beautiful theory that explained why a and 5 are universal (Feigenbaum 1979). He borrowed the idea of renormalization from statistical physics, and thereby found an analogy between a, 5 and the universal exponents observed in experiments on second-order phase transitions in magnets, fluids, and other physical systems (Ma 1976). In Section 10.7, we give a brief look at this renormalization theory. 

First, let us see what we can say about stability for the inviscid fluid. The key is to note that (12-128) and (12 129) are problems of the so-called Sturm Liouville type. This means that we can characterize the sign of the growth-rate factor a2 based on the sign of F. Before drawing any conclusions, it may be useful to briefly review the general Sturm-Liouville theory. The latter relates to the properties of the general second order ODE,... 

---