Dispersion forces in inhomogeneous planarly layered media: A one-dimensional model for effective polarizabilities

Dispersion forces such as van der Waals forces between two microscopic particles, the Casimir-Polder forces between a particle and a macroscopic object, or the Casimir force between two dielectric objects are well studied in vacuum. However, in realistic situations the interacting objects are often embedded in an environmental medium. Such a solvent influences the induced dipole interaction. With the framework of macroscopic quantum electrodynamics, these interactions are mediated via an exchange of virtual photons. Via this method the impact of a homogeneous solvent medium can be expressed as local-field corrections leading to excess polarizabilities which have previously been derived for hard boundary conditions. In order to develop a more realistic description, we investigate a one-dimensional analog system illustrating the influence of a continuous dielectric profile.

Figure 1

Sketch of the considered setups for the spherical problem and the one-dimensional analogon: (a) two particles embedded in a medium creating an Onsager's real cavity with inhomogeneous dielectric profile; (b) one-dimensional analogon with planarly inhomogeneous profile; (c) two spherical nanoparticles embedded in a medium with an inhomogeneous cavity; (d) the corresponding one-dimensional problem with two dielectric plates of finite thickness embedded in a medium with inhomogeneous profile.

Figure 2

Sketch of the corresponding one-dimensional dielectric profiles for (a) the van der Waals force, where two particles are separated by a distance $l$ and embedded in a continuous dielectric cavity (red curve) and (b) the Casimir force, where the two particles have been exchanged by slabs of thicknesses $d_2$. The third picture (c) illustrates the corresponding cases with hard boundaries, which is for the van der Waals force a five-layer system labeled with capital roman numbers (I–V) and for the Casimir force a nine-layer system (small roman numbers i–xi).

Figure 3

Sketch of the three layer system with a centered medium ${\varepsilon }_1$ surrounded by a right ${\varepsilon }_{2+}$ and left medium ${\varepsilon }_{2-}$ illustrating the considered optical paths from the source point $s$ in the centered layer to the final point $f$ far in the right layer. One possible path is marked in black including the multiple scattering inside the cavity. The multiple scattering between the final point and the outer interface between the right and centered medium will not be considered due to the large separation.

Figure 4

Sketch of the different spatial dependent dielectric functions for the hard boundary (red dotted curve), the linear profile (blue curve), and the nonlinear profile (green dashed curve). The black arrows illustrate the considered reflections which take place for the hard boundary at the cavity radius $R_{\mathrm{C}}$ and for the functional profiles at their ends which are shifted by a distance $a$ compared to the hard-boundary case.

Figure 5

Geometric dependence of the reflection coefficients for both profiles—Thomas-Fermi and linear—with the exact solution of the Riccati (solid blue and dashed black lines) equation and the corresponding approximations (dashed light blue and dotted gray lines).

Figure 6

Spatial density profile of water surrounding a helium atom with the cavity radius $R_{\mathrm{C}}$. Simulated profile enclosed the blue area which is approximated by a Thomas-Fermi distribution (red line) with the corresponding thickness of the inhomogeneous region $2a_{\mathrm{TF}}$ and by a linear profile (gray line) with the thickness $2a_{\mathrm{l}}$.

Figure 7

Relative van der Waals potentials for the cavity model with hard boundaries (blue dotted line), linear boundaries (black solid line), and Thomas-Fermi distributed boundary profiles (green dashed-dotted line) with respect to the vacuum potential. The general reduction of the potential caused by the screening in water (factor $\approx 20$) can be observed (red dashed line).

Figure 8

Relative impact of the cavity boundaries on the Casimir force for a helium plate embedded in water relative to the force in vacuum. Without any cavity corrections (dashed red line), with hard boundaries (doted blue line), with the linear profile (solid black line), and with the Thomas-Fermi distributed profile (dashed-dotted green line).