Microspheres at the surface of liquid are widely used now for visualization of wave and vortex motion [1, 2]. The experiments of this kind had been performed recently to study of turbulence at the surface of liquid helium [3]. That’s why it is of interest to consider the corrections to a classic Archimedes' principle, because while the size of a particle floating at the surface decreases, the forces of surface tension and molecular interaction start to play a significant role. 

We study the deviations from Archimedes' principle for spherical particles made of molecule hydrogen near the surface of liquid He⁴. Classic Archimedes' principle takes place if particle radius $R_0$ is greater than capillary length of helium $L_{k} \approx $ 500 µm and the height $h_+$  of the part of the particle above He is proportional to  $R_0$ . Over the range of $30 <R_0 <500$ µm Archimedes' force is suppressed by the force of surface tension and  $h_{+}  \sim R^{3}_{0} / L^{2}_{k}$.  When $R_0<30$µm, the particle is situated under the surface of liquid helium. In this case Archimedes' force competes with Casimir force which repels the particle from the surface to the depth of liquid. The distance from the particle to the surface $h_{-} \sim R^{5/3}_c / R^{2/3}_0$ if  $R_0>R_c...R_c$  can be expressed as  $R_c \approx (\frac {\hbar c}{\rho g}) \approx $ 1µm, $\hbar $ is Planck's constant, c is speed of light, $\rho $ is helium density. For the very small particles ( $R_0<R_c)$  $h_{-}$does not depend on their size: $h_{-}$=$R_c$.

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2.  S. V. Filatov,  D. A. Khramov,   A. A. Levchenko, JETP Letters, 106(5), 330 (2017).

3. A. A. Levchenko, L. P. Mezhov-Deglin, A. A. Pel’menev, JETP Letters, 106(4), 252 (2017).

4. E. V. Lebedeva, A. M. Dyugaev , and P. D. Grigoriev, JETP, 110(4), 693 (2010).

5. A. M. Dyugaev,  P. D. Grigoriev, and E. V. Lebedeva, JETP Letters, 89(3), 145 (2009).

A.M. Dyugaev,  E.V. Lebedeva,

JETP Letters, 106 issue 12, 2017