Density profiles in a classical Coulomb fluid near a dielectric wall. II. Weak-coupling systematic expansions

In the framework of the grand-canonical ensemble of statistical mechanics, we give an exact diagrammatic representation of the density profiles in a classical multicomponent plasma near a dielectric wall. By a reorganization of Mayer diagrams for the fugacity expansions of the densities, we exhibit how the long-range of both the self-energy and pair interaction are exponentially screened at large distances from the wall. However, the self-energy due to Coulomb interaction with images still diverges in the vicinity of the dielectric wall and the variation of the density is drastically different at short or large distances from the wall. This variation is involved in the inhomogeneous Debye-H\"uckel equation obeyed by the screened pair potential. Then the main difficulty lies in the determination of the latter potential at every distance. We solve this problem by devising a systematic expansion with respect to the ratio of the fundamental length scales involved in the two coulombic effects at stake. (The application of this method to a plasma confined between two ideally conducting plates and to a quantum plasma will be presented elsewhere). As a result we derive the exact analytical perturbative expressions for the density profiles up to first order in the coupling between charges. The mean-field approach displayed in Paper I is then justified.Comment: 37 pages, 5 figure

Density profiles in a classical Coulomb fluid near a dielectric wall. I. Mean-field scheme

The equilibrium density profiles in a classical multicomponent plasma near a hard wall made with a dielectric material characterized by a relative dielectric constant \ew are studied from the first Born-Green-Yvon equation combined with Poisson equation in a regime where Coulomb coupling is weak inside the fluid. In order to prevent the collapse between charges with opposite signs or between each charge and its dielectric image inside the wall when \ew >1, hard-core repulsions are added to the Coulomb pair interaction. The charge-image interaction cannot be treated perturbatively and the density profiles vary very fast in the vicinity of the wall when \ew \neq 1. The formal solution of the associated inhomogeneous Debye-H\"uckel equations will be given in Paper II, together with a systematic fugacity expansion which allows to retrieve the results obtained from the truncated \bgy hierarchy. In the present paper the exact density profiles are calculated analytically up to first order in the coupling parameter. The expressions show the interplay between three effects~: the geometric repulsion from the impenetrable wall; the electrostatic effective attraction (\ew >1) or repulsion (\ew <1) due to its dielectric response; and the Coulomb interaction between each charge and the potential drop created by the electric layer which appears as soon as the system is not symmetric. We exhibit how the charge density profile evolves between a structure with two oppositely-charged layers and a three-layer organization when \ew varies. (The case of two ideally conducting walls will be displayed elsewhere)Comment: 32 pages, 11 figure

Nonlinear evolution of a morphological instability in a strained epitaxial film

A strained epitaxial film deposited on a deformable substrate undergoes a morphological instability relaxing the elastic energy by surface diffusion. The nonlinear and nonlocal dynamical equations of such films with wetting interactions are derived and solved numerically in two and three dimensions. Above some critical thickness, the surface evolves towards an array of islands separated by a wetting layer. The island chemical potential decreases with its volume, so that the system experiences a non-interrupted coarsening described by power laws with a marked dimension dependence.Comment: 4 pages, 6 figure

Criticality in Charge-asymmetric Hard-sphere Ionic Fluids

Phase separation and criticality are analyzed in $z$:1 charge-asymmetric ionic fluids of equisized hard spheres by generalizing the Debye-H\"{u}ckel approach combined with ionic association, cluster solvation by charged ions, and hard-core interactions, following lines developed by Fisher and Levin (1993, 1996) for the 1:1 case (i.e., the restricted primitive model). Explicit analytical calculations for 2:1 and 3:1 systems account for ionic association into dimers, trimers, and tetramers and subsequent multipolar cluster solvation. The reduced critical temperatures, $T_c^*$ (normalized by $z$), \textit{decrease} with charge asymmetry, while the critical densities \textit{increase} rapidly with $z$. The results compare favorably with simulations and represent a distinct improvement over all current theories such as the MSA, SPB, etc. For $z$$\ne$1, the interphase Galvani (or absolute electrostatic) potential difference, $\Delta \phi(T)$, between coexisting liquid and vapor phases is calculated and found to vanish as $|T-T_c|^\beta$ when $T\to T_c-$ with, since our approximations are classical, $\beta={1/2}$. Above $T_c$, the compressibility maxima and so-called $k$-inflection loci (which aid the fast and accurate determination of the critical parameters) are found to exhibit a strong $z$-dependence.Comment: 25 pages, 14 figures; last update with typos corrected and some added reference

Density profiles in a quantum Coulomb fluid near a hard wall

Equilibrium particle densities near a hard wall are studied for a quantum fluid made of point charges which interact via Coulomb potential without any regularization. In the framework of the grand-canonical ensemble, we use an equivalence with a classical system of loops with random shapes, based on the Feynman-Kac path-integral representation of the quantum Gibbs factor. After systematic resummations of Coulomb divergences in the Mayer fugacity expansions of loop densities, there appears a screened potential $\phi$. It obeys an inhomogeneous Debye-H\"uckel equation with an effective screening length which depends on the distance from the wall. The formal solution for $\phi$ can be expanded in powers of the ratios of the de Broglie thermal wavelengths \laa's of each species $\alpha$ and the limit of the screening length far away from the wall. In a regime of low degeneracy and weak coupling, exact analytical density profiles are calculated at first order in two independent parameters. Because of the vanishing of wave-functions close to the wall, density profiles vanish gaussianly fast in the vicinity of the wall over distances \laa's, with an essential singularity in Planck constant $\hbar$. When species have different masses, this effect is equivalent to the appearance of a quantum surface charge localized on the wall and proportional to $\hbar$ at leading order. Then, density profiles, as well as the electrostatic potential drop created by the charge-density profile, also involve a term linear in $\hbar$ andwhich decays exponentially fast over the classical Debye screening length \xid. The corresponding contribution to the global surface charge exactly compensates the charge in the very vicinity of the surface, so that the net electric field vanishes in the bulk, as it should.Comment: 48 pages, 6 figure

PHYSIQUE STATIQUE DES FLUIDES COULOMBIENS CLASSIQUES ET QUANTIQUES AU VOISINAGE D'UNE PAROI

ORSAY-PARIS 11-BU Sciences (914712101) / SudocSudocFranceF

Van der Waals Heteroepitaxy of Air-Stable Quasi-Free-Standing Silicene Layers on CVD Epitaxial Graphene/6H-SiC

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