Casimir microsphere diclusters and three-body effects in fluids

Our previous paper [Phys. Rev. Lett. 104, 060401 (2010)] predicted that Casimir forces induced by the material-dispersion properties of certain dielectrics can give rise to stable configurations of objects. This phenomenon was illustrated via a dicluster configuration of nontouching objects consisting of two spheres immersed in a fluid and suspended against gravity above a plate. Here, we examine these predictions from the perspective of a practical experiment and consider the influence of nonadditive, three-body, and nonzero-temperature effects on the stability of the two spheres. We conclude that the presence of Brownian motion reduces the set of experimentally realizable silicon-teflon spherical diclusters to those consisting of layered microspheres, such as the hollow core (spherical shells) considered here.

Figure 1

Schematic of two-sphere dicluster geometry consisting of two dielectric spheres of radii $R_1$ and $R_2$ separated by a center-center distance $d$ from each other, and suspended by heights $h_1$ and $h_2$, respectively, above a dielectric plate.

Figure 2

Equilibrium separation $d_e(h)/d_e(\infty )$ between two $R=25$ nm spheres suspended in ethanol as a function of their surface-surface separation $h$ from a plate (and normalized by the equilibrium separation for the case of two isolated spheres, i.e., $h=\infty$). $d_e$ is plotted for various material combinations, denoted by the designation sphere-sphere-plate, e.g., a PS and silicon sphere suspended above a gold plate is denoted as PS-Si-Au. Solid (dashed) lines correspond to stable (unstable) equilibria. (In the case of a gold plate, the spheres are chosen to have $R=50$ nm.) The inset shows $d_e$ (in units of nm) for the case of two PS and silicon spheres ($R=57$ nm) above a gold plate.

Figure 3

Equilibrium separation $d_e(\infty )$ (units of nm) between a Si sphere and either a teflon (Tef) or polystyrene (PS) sphere immersed in ethanol as a function of their equivalent radii $R$. Solid (dashed) lines denote stable (unstable) equilibria.

Figure 4

Center-surface $L_e$ (solid lines) and surface-surface $h_e$ (dotted lines) equilibrium separation (units of nm) of a teflon (red) or Si hollowed sphere (shown in the inset) suspended in ethanol above a gold plate, as a function of radius $R$ (units of nm). The equilibria are plotted for different values of the fill fraction $\alpha$, defined as the ratio of the spherical-shell thickness over the radius of the sphere. Solid (dashed) lines correspond to stable (unstable) equilibria.

Figure 5

Geometry of a hollow air-filled core sphere suspended above a layered plate with layer thickness $H$. Explicitly shown is the thickness dimension as a function of $\alpha$.

Figure 6

Surface-surface equilibrium height $h_e$ (units of nm) for the hollowed-sphere geometry of Fig. 5, consisting of either a Si (top) or teflon (bottom) hollowed sphere (fill fraction $\alpha$) suspended in ethanol above an $H=15$ nm ITO layered gold plate, as a function of sphere radius $R$ (in units of $\mu$m). Solid (dashed) lines correspond to stable (unstable) equilibria. $h_e$ is plotted for different values of $\alpha$, denoted in the figure. The top inset plots the center-surface separation $L_e$ (in units of nm) as a function of $R$ of a hollowed teflon (red lines) and Si (blue lines) sphere suspended again above a gold plate, for $\alpha =0.14$ (0.142). The lower inset shows $h_e$ for both teflon and Si spheres for $R\in [8.6,10]\hspace{0.5em}\mu$m.

Figure 7

Average $\langle h\rangle$ (thick lines) and equilibrium $h_e$ (thin lines) height (in units of nm) of a hollowed teflon (blue lines) and Si (red lines) sphere suspended above an $H=15$ nm ITO layered gold plate, for $\alpha =0.142$ (0.13), as a function of sphere radius $R$ (in units of $\mu$m). Solid (dashed) lines correspond to stable (unstable) equilibria. The red (blue) shaded regions indicate positions where the teflon (Si) spheres are found with $95\%$ probability. The inset shows $\langle d\rangle$ (thick line) and $d_e$ (thin line) separations as a function of their radius for two equal radii teflon Si spheres. The gray shaded region indicates the separations in which the teflon and Si spheres are found with $95\%$ probability.

Figure 8

Average $\langle h\rangle$ (thick line) and equilibrium $h_e$ (thin line) height (in units of nm) of a hollowed Si (top) and teflon (bottom) sphere of radii $R=10$ ($9.915)\hspace{0.5em}\mu$m suspended in ethanol above an $H=15$ nm ITO layered gold plate, as a function of fill fraction $\alpha$ (indicated in Fig. 5). Shaded regions indicate $h$ positions where the Si (teflon) spheres are found with $95\%$ probability. Solid (dashed) lines indicate stable (unstable) equilibria. For reference, we state the equilibrium $d_e$ and average $\langle d\rangle$ horizontal separations between $R=10$ ($9.915)\hspace{0.5em}\mu$m Si (Tef) spheres in the top figure.

Figure 9

Energy barrier $\Delta /\mathit{kT}$ of a hollowed teflon sphere suspended in ethanol above an $H=15$ nm ITO layered gold plate at $T=300$ K, as a function of sphere radius $R$ (in units of $\mu$m) and for different values of fill fraction $\alpha$. The inset shows the energy landscape $U/kT$ as a function of the surface-surface height $h$ (units of nm) for a teflon sphere of radius $R=10\hspace{0.5em}\mu \text{m}$ with a fill fraction of $\alpha =0.142$.