Nonuniform Fluid Research Articles

Extension of the square-gradient theory to fourth order

In the square-gradient theory, the Helmholtz free energy density of a nonuniform fluid is approximated by that of a uniform fluid plus a term proportional to the square of the density gradient. Presented here is the extension of this theory (and the corresponding theory of the chemical potential) to fourth order in the gradients. The new results can be applied to study the pair correlation function and interfacial density profile in a fluid near its critical point.

On the validity of certain integro-differential equations for the density—orientation profile of molecular fluid interfaces

The status of some recently proposed integro-differential equations for the density—orientation profile, p( r 1, ω 1), of non-uniform molecular fluids is examined. Two of the equations derived by Gubbins, which involve derivatives with respect to angular coordinates ω 1, are shown to be incorrect and corrected versions are proposed. The implications of the results for pairwise correlations and for the surface tension of the liquid—gas interface are discussed.

Percus-Yevick equation for an inhomogeneous fluid: Critical region

The critical region of a locally nonuniform fluid with gaussian density inhomogeneity is investigated. Nonclassical behaviour is found; critical exponents agree very well with those obtained from the perturbation theory of liquids. The action of an external field shifts the critical region and weakens the critical behaviour. Discussed effects could be verified experimentally.

A perturbation approach to the density functional theory of adsorption

A modified treatment of an exact density functional theory for non-uniform fluids due to Saam and Ebner is presented and a perturbation method of solution is proposed which avoids the need for a local density approximation. For hard-core fluids near a hard wall the equation for the equilibrium density reduces tothe wall-particle version of the RHNC approximation.

Applications of Noether's theorem to inhomogeneous fluids

The implications of Noether's theorem in the van der Walls theory of nonuniform fluids are examined for a two-phase system with a planar interface, for a three-phase fluid with a line of contact, and for a two-phase fluid with a pair of coexisting interface structures. In each case the thermodynamic consequences of the resulting conservation laws are examined.

Interfacial hydrodynamics: A microscopic approach

Linearized hydrodynamic equations for a nonuniform anisotropic fluid are obtained from the exact Mori–Zwanzig equations for the conserved densities. In the particular case of a two-phase system with a planar equilibrium interface, these equations can be reduced to the ordinary hydrodynamic equations inside each bulk phase and to surface hydrodynamic equations for the interfacial layer. Surface transport coefficients and surface thermodynamic parameters are hereby obtained as Gibbs surface excess values. All the known phenomenological equations can be recovered by suitable approximations. Various correction terms to the phenomenological results, including Laplace’s formula, are found.

Applications of singlet density-direct correlation function equations to surface tension of multicomponent fluids and to the theory of freezing

Sokolowski [1] has recently written down equations relating singlet densities to direct correlation functions in a non-uniform multicomponent fluid. These same equations, derived from translational invariance requirements, have been used previously by Bhatia et al. [2] to calculate the surface tension of multicomponent fluids. This application is briefly discussed, as well as a further, approximate, use of the same equations in the theory of freezing of liquid binary alloys and of ionic melts.

Statistical mechanics of a nonuniform fluid with long-range attractions

A recursive method is described for generating the Helmholtz free energy and paircorrelation function of a nonuniform fluid, in powers of the Kac inverse-range parameter $\ensuremath{\gamma}$. The results agree with earlier calculations using graphical and functional-integral methods. The ordering of the free energy is functionally consistent with a similar ordering of the density profile. The lowest-order or "van der Waals" theory is equivalent to a mean-field average of the attractive intermolecular energies and use of local thermodynamics in a reference hard-sphere fluid. The next level of approximation is the "random-phase approximation," which is shown to produce logarithmic anomalies in the free energy and density profile similar to those expected on the basis of capillary-wave theory. The occurrence of these anomalies results from the existence of a longwavelength divergence in the transverse Fourier transform of the lowest-order paircorrelation function.

Solvation forces in charged fluids

A formalism based on linear response theory is used to obtain an expression for the free energy of a non-uniform charged fluid in terms of the local ion number density and the bulk direct correlation functions. When the fluid is a restricted primitive model electrolyte the free energy splits into two independent parts, the minimization of which leads to expressions for the equilibrium charge and density distributions. From the free energy an expression for the force between two thick plates immersed in an electrolyte is obtained. In the limit of point ions, the expressions we obtain reduce to those of the Debye-Huckel theory of electrolytes. The equations are solved numerically and at low bulk electrolyte concentrations the monotonically decaying repulsive force of the classic Verwey and Overbeek results is found. But at higher concentrations and larger inverse Debye screening lengths the force displays pronounced oscillations. Correspondingly, the electric potential displays oscillations which have conse...

The effect of nonuniform fluid flow distribution on the thermal performance of solar collector

In the design of solar collector, every effort is made to provide an uniform fluid flow distribution among the flow channels of the collector. Economical and space considerations, however, may make the uniform fluid flow distribution through the solar collector unrealistic. If the fluid flow distribution is not uniform, the thermal performance of the solar collector may deteriorate. A numerical method is developed for determination of the deterioration of thermal performance of solar collector due to this flow nonuniformity effect. Using several typical fluid flow maldistribution models, the collector efficiency and its deterioration are determined for various collector design and operating conditions.

Three-wave coupling coefficients for non-uniform plasmas

Resonant three-wave interaction in a spatially non-uniform magnetized fluid plasma is considered. We calculate the coupling coefficientsVjfor wave propagation in arbitrary directions and demonstrate that new terms, which appear inVjowing to the presence of density gradients, can dominate actual wave coupling processes.

On the relationship between the density functional formalism and the potential distribution theory for nonuniform fluids

It is shown that the variational principle for the grand potential of a nonuniform fluid as a functional of the singlet density yields the potential distribution theory for the equilibrium density. We derive the explicit form that the functional takes for a system of hard rods, and propose an approximate one for hard spheres. Attractive interactions are also considered in mean-field approximation. In all cases the pair direct correlation function of the nonuniform system is obtained and the density gradient expansion of the free energy is investigated.

Propagation of Density Fluctuations in Nonuniform Fluids: Simple Models

As a first step toward the understanding of Rayleigh-Brillouin scattering from a fluid with a temperature gradient, we analyze the initial value problem for certain prototype one-dimensional nonuniform systems. For sufficiently short times and localized initial pulses, it is not necessary to impose actual physical boundaries on the linearly nonuniform system. By a straightforward, though unconventional, application of Fourier and Laplace transform methods, explicit and physically reasonable solutions are constructed for the propagation of fluctuation pulses in space and time according to the nonuniform wave and diffusion equations and a special case of the nonuniform damped wave equation. The analog of the dynamic structure factor is also constructed for the latter two cases.

Model grand potential for a nonuniform classical fluid

Classical equilibrium statistical mechanics of one-dimensional hard cores in an arbitrary external field is reviewed. The grand canonical potential is shown to be an extension of local thermodynamics in which suitable averages of density and pressure are used. A general three-dimensional system with arbitrary internal and external potentials is modeled in corresponding form, and the external potential represented as a functional of the density. Comparison with assumed bulk data permits evaluation of the required averages. It is shown that the wall equation of state for a wall-bounded fluid is satisfied exactly. Application is also made to the self-maintained density profile of the two-phase fluid at a first order phase transition.

Linear and non-linear theories of solvation forces in fluids

From an exact expression for the free energy of a non-uniform classical fluid, due to Saam and Ebner, a closure is used to develop a non-linear theory for the density and solvation force between two planar walls. In the linear limit these expressions reduce to ones used successfully elsewhere. Numerical solution of the equations for a hard sphere fluid shows that while the density profiles predicted by the two theories are markedly different, the solvation forces are similar.

Theory of excess-electron states in classical rare-gas fluids

Localized excess-electron states are investigated using the density-functional formalism previously developed by Ebner, Saam, and Stroud for nonuniform classical fluids. The electron-atom interactions used contain a short-range effective core repulsion and a long-range polarization attraction. The interatomic interaction is a 6-12 potential. Some results are in agreement with previous density-functional calculations by Ebner and Punyanitya, who use a contact potential for the electron-atom interaction, in that no localized states are found in dilute helium and neon gases, while localized states are found in dense helium gas and liquid neon close to the liquid-gas coexistence curve. However, our present calculations disagree with the earlier ones in that they predict the localized states in liquid neon to be stable even at pressures much larger than the saturated vapor pressure, a consequence, we believe, of our inclusion of the polarization potential in the electron-atom interaction. We also find that localized states in the form of short-lived droplets are possible in argon and xenon gases close to the critical point. This finding is consistent with recent observations by Freeman and Huang of electron mobilities in these gases.

Rigorous interface properties for the van der Waals fluid mixture

A statistical mechanical treatment for the van der Waals fluid mixture is developed, based on Widom’s potential-distribution theory for nonuniform fluids. For the specific choice of Kac pair interactions, in the van der Waals limit, a set of coupled second-order differential equations determine the density profiles. The solution of these equations and the determination of the interfacial tension are discussed in terms of a mechanical analogy.

Structure of non-uniform molecular fluids: Integrodifferential equations for the density-orientation profile

A family of integrodifferential equations are derived for the density -orientation profile, p(z1ω1), in a non-uniform molecular fluid. Three of these equations are generalizations of well-known results for simple (spherical molecule) fluids, while the remaining three apply only to molecular fluids. Applications to the gas-liquid interface are briefly discussed.

Film formation on a weakly attractive substrate within the lattice-gas model

Adsorption of a dense film on a weakly attractive substrate is studied using a lattice-gas model in the mean-field approximation. The model and method of solution are much the same as those employed by de Oliveira and Griffiths. In the present theory, the particles of the gas interact via either a Lennard-Jones (6-12) potential or a nearest-neighbor potential, and a gas particle interacts with the substrate through a (9-3) potential. For sufficiently weak substrate potentials no film is found. For sufficiently strong ones a film forms and grows with increasing vapor pressure or chemical potential either continuously or in one-atomic-layer steps as found by de Oliveira and Griffiths. For potentials of intermediate strength, there is a discontinuity in the film thickness from a partial layer to an arbitrarily large number of layers, depending on the temperature, as the vapor pressure is increased. The phase diagram in these cases is strikingly similar to that found by Ebner and Saam from a density-functional theory of nonuniform fluids; there is, however, a sizable difference in the substrate potential strengths at which the large discontinuity in film thickness appears.

On the summation of the virial expansion for the local density of a gas in contact with a solid

We investigate the density profile of a fluid in contact with a wall. Our analysis is based on the summation of the virial expansion for the local density and the resulting integral equation for the density profile involves the two particle direct correlation function of a non-uniform fluid.