Spreading of Normal Liquid Helium Drops

We have used video imaging and interferometric techniques to investigate the dynamics of spreading of drops of ${}^4\mathrm{He}$ on a solid surface for temperatures ranging from 5.2 K (near the critical point) to 2.2 K (near $T_{\lambda }$). After an initial transient, the drops become pancake-shaped with a radius that grows as $R\left(t\right)\approx t^{\alpha },$ with $\alpha =0.149\pm 0.002$. The drops eventually begin to shrink due to evaporation driven by gravitational and curvature effects, which limits their lifetime to about 1000 s. Although helium completely wets the substrate, and the spreading takes place over a pre-existing adsorbed film, a distinct contact line with a contact angle of order one degree is visible throughout this process.

Figure 1

Schematic diagram of cryogenic and optical apparatus. Drops were delivered from a capillary tube approximately 1 cm above the substrate, which was usually a sapphire plate. The spreading drop was illuminated with collimated and filtered LED light, which was shone at a shallow angle for the edge-on view, or reflected from a 45 degree mirror to produce normal incident light for the bottom view. The level of the bulk liquid in the cell was approximately 1 cm below the top of the substrate.

Figure 2

Drop seen from the side, approximately 2 s after impact. Temperature is 2.5 K. Interference lines show the changing thickness of the drop and are visible due to the slight angle of the imaging system.

Figure 3

Effect of heat input on drop lifetime and evolution. Power is applied to the sapphire substrate via an ohmic heater epoxied to the side. (a) Drop duration as a function of heat input for various temperatures. At low temperatures, superfluid drops, shown in blue, have short lifetimes that are independent of heater power. Superfluid drops near $T_{\lambda }$, shown in black, have a longer lifetime, which is nevertheless independent of power. The lifetime of drops in the normal state, shown in red, ranges from $\approx 300–1000$ s at nominally zero power to a few seconds with 0.1 mW of power. (b) Drop spreading radius as a function of time for several values of input power. The radius grows roughly proportional to $t^{1/7}$ until evaporation dominates and the drop starts to shrink. Microwatts of power were sufficient to noticeably increase the evaporation rate.

Figure 4

Schematic diagram of a drop on a sapphire substrate. The sapphire and the vapor above the drop are at cell temperature $T$. The pressure in the vapor is $P_{\text{sat}}\left(T\right)$ at the bulk liquid-vapor interface, which is a distance $H\approx 1\mathrm{cm}$ below the drop; the vapor pressure above the drop is lower than $P_{\text{sat}}\left(T\right)$ by an amount ${\rho }_{v}gH$. The pressure in the drop is larger than $P_{\text{sat}}\left(T\right)$ due to the Laplace pressure ${\sigma }_{\text{lv}}H/R^2$. These pressure differences drive an evaporative flux out of the drop, which lowers the drop temperature $T_d$.

Figure 5

Drop profile constructed from the interference patterns as a function of time. The horizontal and vertical scales differ by almost a factor of 1000, so the physical drops are exceedingly flat.

Figure 6

Drop radius as a function of time for drops at $T=2.5$ K. The best-fit spreading exponent of $0.149\pm 0.002$ using the data with $4\text{s}<t<140\text{s}$ is shown as the black line. The best-fit exponent is slightly greater than 1/7 = 0.143.

Figure 7

Plot of ${\chi }^2$ for fits of the form $R\left(t\right)\sim t^{\alpha }$ to the data of Fig. 6 as a function of $\alpha$.

Figure 8

Numerical solutions for the evolution of the drop profile, showing the drop height in mm as a function of the footprint radius $R$ in mm for various values of the the time $t$ in seconds.