Fluctuation-Induced Interaction between Randomly Charged Dielectrics

Monopolar charge disorder effects are studied in the context of fluctuation-induced interactions between neutral dielectric slabs. It is shown that quenched bulk charge disorder gives rise to an additive contribution to the net interaction force which decays as the inverse distance between the slabs and may thus completely mask the standard Casimir–van der Waals force at large separations. By contrast, annealed (bulk or surface) charge disorder leads to a net interaction force whose large-distance behavior agrees with the universal Casimir force between ideal conductors, which scales as the inverse cubic distance, and the dielectric properties enter only in the subleading corrections.

Figure 1

A typical experimental set up (bottom) is modeled with a plane-parallel system (top) of two dielectric slabs (half-spaces) of dielectric constant ${\epsilon }_p$ interacting across a medium of dielectric constant ${\epsilon }_m$. The charge distribution in the bulk of the slabs and/or on the two bounding surfaces at $z=\pm D/2$ has a disordered component (shown schematically by small light and dark patches) with zero mean but finite variance, and may be either quenched or annealed in nature.

Figure 2

(a) Ratio of the total force (8) to the zero-frequency vdW force (6) between net-neutral dielectric half-spaces in vacuum (${\epsilon }_m=1$) bearing quenched monopolar charge disorder for fixed bulk and surface disorder variances $g_b=5\times {10}^{-8}{\mathrm{nm}}^{-3}$, $g_s=g_b^{2/3}$ and different dielectric constants ${\epsilon }_p=2$, 5, 10, 20, 40, 80, 100 (dashed curves from top). (b) Magnitude of the rescaled total force (8) in the quenched case as a function of the rescaled distance for fixed ${\epsilon }_p=10$, $g_s=0$ and various bulk disorder variances $g_b={10}^{-6}$, ${10}^{-7}$, ${10}^{-8}$, ${10}^{-9}$, ${10}^{-10}$, ${10}^{-11}{\mathrm{nm}}^{-3}$ (dashed curves from top). Solid curve is the pure vdW force (6). Inset is the ratio of the total force to the vdW force (6) for the same range of $D$. (c) is the same as (b) but for annealed disorder [top panel, from Eq. (12)]. Annealed curves stay close to one another and are bracketed by the universal force (16) for strong disorder [top solid line, labeled by $\zeta (3)$] and the nonuniversal vdW force (6) for no disorder [bottom solid line, labeled by ${\mathrm{Li}}_3({\Delta }^2)$] as seen more clearly from the force ratio shown in the bottom panel. (b) and (c) are plotted in log-log scale.