Casimir-Polder force between anisotropic nanoparticles and gently curved surfaces

The Casimir-Polder interaction between an anisotropic particle and a surface is orientation dependent. We study novel orientational effects that arise due to curvature of the surface for distances much smaller than the radii of curvature by employing a derivative expansion. For nanoparticles we derive a general short distance expansion of the interaction potential in terms of their dipolar polarizabilities. Explicit results are presented for nano-spheroids made of ${\mathrm{SiO}}_2$ and gold, both at zero and at finite temperatures. The preferred orientation of the particle is strongly dependent on curvature, temperature, as well as material properties.

Figure 1

Parametrization the configuration of a nano-spheroid near a gently curved surface: Solid curves on the surface $S$ indicate the principal directions at $P$ with local radii of curvature $R_1$ and $R_2$. Positive (negative) $R_j$ correspond to a surface that curves away from (towards) the particle.

Figure 2

Stable orientations of the particle’s symmetry axis as function of the coefficients $B$ and $D=C(d/R_1-d/R_2)$ in Eq. (11).

Figure 3

Typical surface shapes for limiting cases corresponding to combinations of $R_1=\pm 1$, $R_2=\pm 3$ and $R_2\to \infty$ (arbitrary units). The coordinate frames indicate the position and orientation of the particle.

Figure 4

Stability diagram for a ${\text{SiO}}_2$ oblate spheroid (“pancake”) with $n_3=0.7$ and $R_1=1000\mu \mathrm{m}$ at $T=0\mathrm{K}$.

Figure 5

Stability diagram for a ${\mathrm{SiO}}_2$ prolate spheroid (“needle”) with $n_3=0.2$ and $R_1=100\mu \mathrm{m}$ at $T=0\mathrm{K}$.

Figure 6

Stability diagram for a ${\mathrm{SiO}}_2$ oblate spheroid (“pancake”) with $n_3=0.7$ and $R_1=1000\mu \mathrm{m}$ at $T=300\mathrm{K}$.

Figure 7

Stability diagram for a gold oblate spheroid (“pancake”) with $n_3=0.7$ at $T=0\mathrm{K}$.

Figure 8

Stability diagram for a gold prolate spheroid (“needle”) with $n_3=0.2$ at $T=0\mathrm{K}$.

Figure 9

Stability diagram for a gold oblate spheroid (“pancake”) with $n_3=0.7$ and $R_1=20\mu \mathrm{m}$ at $T=300\mathrm{K}$.

Figure 10

Stability diagram for a gold prolate spheroid (“needle”) with $n_3=0.2$ and $R_1=20\mu \mathrm{m}$ at $T=300\mathrm{K}$.