The contact line dynamics over a sinking solid sphere are investigated in comparison to classical spreading theories. Experimentally, high-speed imaging systems with optical light or x-ray illumination are employed to accurately measure the spreading motion and dynamic contact angle of the contact line. Millimetric spheres are controlled to descend with a constant speed ranging from 7.3 × 10^{–5} to 0.79 m/s. We observed three different spreading stages over a sinking sphere, which depends on the contact line velocity and contact angle. These stages consistently showed the characteristics of capillarity-driven spreading as the contact line spreads faster with a higher contact angle. The contact line velocity is observed to follow a classical capillary-viscous model at a high Ohnesorge number (>0.02). For the cases with a relatively low Ohnesorge number (<0.02), the contact line velocity is significantly lower than the speed predicted by the capillary-viscous balance. This indicates the existence of an additional opposing force (inertia) for a decreasing Ohnesorge number. The capillary-inertial balance is only observed at the very beginning of the capillary rise, in which the maximum velocity is independent of the sphere's sinking speed. Additionally, we observed the linear relationship between the contact line velocity and the sphere sinking speed during the second stage, which represents capillary adjustment by the dynamic contact angle.

Dynamics of the triple contact line between a solid and two immiscible fluids have brought up interesting scientific questions and inspired many engineering applications.^{1–5} Traditionally, the spreading behavior of the triple contact line is described as a result of the balance between capillarity and resistant forces in the vicinity of the contact line.^{6} Hoffman in 1975 performed pioneering experimental work and established a relationship between the spreading speed (*U _{c}*) and the dynamic contact angle (

*θ*)

^{7}and later provided the molecular explanation for the experimental data.

^{8}In 1986, de Gennes derived a mathematical relation as $Uc\u221d\sigma \theta (\theta 2\u2212\theta s2)/\mu $ for small

*θ*, which is referred to as the Hoffman-de Gennes equation,

^{9,10}where

*σ*is the surface tension,

*μ*is the liquid viscosity, and

*θ*is the static contact angle. This capillary-viscous model can also be employed to describe the spreading speed of the contact line as a function of time (

_{s}*t*). When spreading occurs axisymmetrically, the spreading speed follows $Uc\u221dt\u22129/10$, known as Tanner's law.

^{11}Theoretically, Hoffman-de Gennes and Tanner's equations are interchangeable in the case of a spreading droplet with a simple geometry.

^{12}Even though these equations are derived only for small

*θ*, the equations are experimentally proven to be valid even up to $\theta \u2248150\xb0$ within minor error.

^{1,13,14}It would suggest that the effective region of a small angle exists to endure dominant viscous dissipation.

^{15}A similar solution, 9$Caln(\alpha lM/lm)=\theta 3\u2212\theta s3$, was derived for small $Ca(=\mu Uc/\sigma \u226a1)$ from the matched asymptotic expansion between micro-, meso-, and macroscopic regions. This is known as the Cox-Voinov equation,

^{1,16,17}where

*l*and

_{M}*l*are macroscopic and microscopic length scales, respectively, and

_{m}*α*is a constant.

Besides the capillary-viscous balance, different balances have been introduced to describe the spreading dynamics in different situations.^{6} For example, in the case of spreading for a large liquid drop ($a\u226blc=\sigma /(\rho lg)$, where *l _{c}* is the capillary length,

*a*is the initial drop radius and

*ρ*is the liquid density), a different power law has been observed as $Uc\u221dt\u22127/8$ from the balance between gravity and viscosity.

_{l}^{18}In the very initial stage of a liquid drop spreading, the fluid inertia, rather than viscosity, balances with the capillary force, thereby showing $Uc\u221dt\u22121/2$.

^{19}

The contact line dynamics of a solid object entering a liquid-air interface have been studied extensively for a plunging plate.^{9,15,16,20,21} While a vertical plate plunges into a liquid bath, the contact line velocity relative to the plate increases with the plunging speed until it reaches a critical speed. This critical speed has been observed to be proportional to the capillary-viscous velocity.^{9,16,20,22} The critical contact line speed is the same as air entrainment velocity as air starts to be dragged by the plate because the contact line cannot keep up with the plunging speed.^{15,23}

Force balances on a contact line have been extensively discussed in the context of the drop impact,^{20,24–26} which should be valid for the contact line on a sinking sphere. In drop-impact dynamics, the Ohnesorge number ($=\mu /\rho \sigma l$) and Weber number ($=\rho U2l/\sigma $) are non-dimensional numbers to characterize the spreading mechanism, where *l* and *U* are the characteristic length and velocity scales, respectively. The Ohnesorge number can be interpreted as the ratio of the inertial time scale ($\rho l3/\sigma $) over the viscous time scale ($\rho l2/\mu $).^{24} Thus, as Oh increases, the dominant resisting force changes from inertia to viscous stress.^{24,25} The dominant driving pressure is known to transition from capillary ($\u223c\sigma $) to impact pressure ($\u223c\rho lU2l$) as the Weber number (We) increases. However, such a transition in spreading dynamics has not been discussed well for the present problem of contact line motion over a sinking sphere. In the present work, we focus on the capillarity-driven mechanism in a range of small and moderate Weber numbers ($=\rho lUs2lc/\sigma $) from 3.8 × 10^{–7} to 4.4 × 10^{1}.

We designed and performed experiments of a sphere falling at a constant speed, *U _{s}*. Either the top or bottom of the sphere was tethered to the linear stage (BiSlide MB10-0150/Velmex, Inc.) as illustrated in Fig. 1. For high-speed x-ray imaging, a polychromatic beam with the 1st harmonic energy near 13 keV was used to illuminate the system to resolve the triple contact line [see Figs. 2(b) and 2(c)]. A 100

*μ*m-thick LuAG:Ce scintillator was employed to convert the x-rays into visible light photons, and a high-speed camera (Fastcam SA1.1/Photron Inc.) recorded the images with a frame rate up to 25 000 hz. To avoid overheating of the scintillator by the intense x-ray beam, shutters were used to allow the x-ray to expose to samples for less than 40 ms.

^{27}However, to observe the overall spreading dynamics as shown in Fig. 2(a), we used optical light illumination. Here, we used a high-speed camera (Fastcam Mini UX100/Photron Inc.) with a frame rate up to 6400 Hz.

We tested four different liquids: [*ρ _{l}* (kg/m

^{3}),

*μ*(g/m $\xb7$ s),

*σ*(mN/m)] = [998, 1.0, 73] for water, [762, 1.2, 22] for ethanol, [965, 9.7, 20] for PDMS 10 cSt, and [965, 48.3, 20] for PDMS 50 cSt. Solid spheres with different materials were used: glass, steel, aluminum, and polyoxymethylene (Delrin). Static contact angles for water were measured by the sessile drop method as $\theta s=57\xb15\xb0$ for glass, $85\xb13\xb0$ for steel, $91\xb13\xb0$ for aluminum, and $101\xb17\xb0$ for Delrin (N = 3 for all materials). The values of

*θ*from the other liquids are nearly zero as they totally wet the solid surface due to its low

_{s}*σ*. We also prepared more hydrophilic glass spheres ($\theta s=24\xb110\xb0$ for water) by applying air-plasma treatment for 3 min using a plasma cleaner (PE-25/Plasma Etch Inc.). The sphere diameter,

*D*, and the sinking velocity,

*U*, varied in the ranges of $0.79\u2264D\u22649.5$ mm and $3.2\xd710\u22124\u2264Us\u22640.79$ m/s, respectively. Detailed experimental conditions are listed in Tables S1–2 in the supplementary material. Using the Mathematica in-house code, the angular location ($\varphi $) of the contact line with respect to the sphere center was tracked over time

_{s}*t*. First, the contact line velocity was calculated by differentiating the position data as $Uc=0.5D(\Delta \varphi /\Delta t)$, which represents the velocity in the moving frame. Then, the dynamic angle was (

*θ*) was measured at the microscale level down to 2

*μ*m/pixel by using the manual tracking module in ImageJ.

In the experiments, we identified three spreading stages based on kinematic behaviors as shown in Fig. 3, where *t* is normalized by the pinch-off time, *t _{p}*. In the first stage, when the contact line position (filled diamonds) is near the bottom of the sphere ($\varphi \u2273\u2212\pi /2$), the contact line spreads rapidly in the beginning but decelerates soon after. Meanwhile, the dynamic contact angle

*θ*(open diamonds) rapidly decreases and reaches a constant contact angle as the sphere sinks. Then, in the second stage near the equator of the sphere ($\varphi \u223c0$), the contact line position changes linearly with

*t*and the dynamic contact angle stays at a terminal value. A similar observation of $\varphi \u221dt$ was experimentally reported for a sinking cylinder; however, the underlying mechanisms were not discussed.

^{28}In the last stage, just before the pinch-off ($\varphi \u2272\pi /2$), both the speed and the angle of the contact line increase with time. This behavior is primarily driven by the pinch-off dynamics in a cylindrical air cavity above the contact line.

^{29}The regions shortly after

*t*/

*t*= 0.25 in blue and 0.65 in red in Fig. 3 have missing data due to an overlap between the contact line and the free surface. Figure 3(b) shows the changes in the normalized contact line velocity of two different Oh numbers. The contact line velocity decreases in the first stage but reaches a terminal speed in the second stage. This terminal velocity is close to the sphere-sinking speed regardless of Oh numbers. A similar trend in the contact line velocity has been observed in a plunging plate,

_{p}^{15,21,23}in which the contact line slows down to the plunging speed by adjusting the contact angle (see Fig. S1 in the supplementary material).

^{21}

As described earlier, a change in the contact line velocity is closely related to a change in the dynamic contact angle, which reveals the characteristics of a capillarity-driven mechanism. The classical Hoffman-de Gennes equation describes the spreading dynamics as a capillary-viscous model in most of the present experimental range of *θ.*^{13–15} Figure 4(a) shows the relationship between the instantaneous capillary number, $\mu Uc(t)/\sigma $, and instantaneous contact angle difference, $\theta (t)[\theta (t)2\u2212\theta s2]$. All symbols are colored by the Ohnesorge number ($Oh=\mu /\rho l\sigma lc$), where the capillary length *l _{c}* is used as the characteristic length scale

^{1,9,15,26}rather than the sphere diameter

*D*.

Figure 4(a) shows that experiments with high Ohnesorge numbers (red symbols) follow the capillary-viscous model. Here, the capillary-viscous model of $Ca=0.013\theta (\theta 2\u2212\theta s2)$ is plotted as a solid line, where the constant prefactor of 0.013 is used based on Hoffman's previous experiments.^{7} The prefactor weakly depends on a spreading geometry and boundary condition logarithmically.^{1,17} This capillary-viscous model is in good agreement with experiments with high Oh (>0.02).^{25} It is worth noting that Fig. 4(a) shows high scattering in the data; however, sub-sampled data less than a certain Weber number reduce the scattering (for details, see Fig. S1 in the supplementary material). It suggests that data become dispersed from a capillary-viscous model due to the effect of impact pressure. Figure 4(b) plots the instantaneous $Uc(t)$ versus *U _{s}*, where data in the first stage (symbols with a black edge) are clearly separated from data in the second stage (symbols without an edge). At the very beginning of the first stage, the contact line is anticipated to move as a balance between capillarity and inertia. This balance predicts a capillary-inertial velocity ($UCI=\sigma /(\rho llc)$) which is independent of a sphere sinking speed. The capillary-inertial velocity is shown as a gray stripe in Fig. 4(b): 0.12–0.16 m/s depending on different liquids we used (see Table S1 in the supplementary material). We observed that data points at the beginning of the first stage started from the capillary-inertial velocity. Similarly, the capillary-inertial velocity was observed only within a short time just after the drop impact.

^{19}When the capillary-inertia rise becomes slower than the falling speed of a sphere as in Fig. 5(b), the sphere sinks before the meniscus rises up above the free surface. After this very early stage, the contact line velocity slows down to less than

*U*, while

_{CI}*θ*decreases. As shown in Fig. 4(a), the contact line velocity with a relatively small Oh is observed to be significantly lower than that predicted in the capillary-viscous model. These data are not governed by simple scaling dynamics as neither inertia nor viscous resistance is negligible. It shares the same conclusion with a previous study, where the capillary and viscous properties were key factors for determining flow characteristics near a sinking solid sphere, even on water.

^{20}Since the driving force spreads both the viscous resistance and inertia, the capillary-viscous model would predict a higher contact line speed. A similar force balance has been observed and discussed in the case of drop dynamics.

^{19,24,25,30}

The linear relation of ‘$Uc\u2248Us$’ in the second stage [Fig. 4(b)] is discussed with the terminal *θ*, which indicates that the contact line velocity reaches the plunging speed by the capillary adjustment. As presented in Fig. 3(a), the contact angle reaches a terminal value as the second stage begins, similar to the case of a plunging plate.^{21} For this second stage, we observed that the contact angle, *θ*, increases with the plunging speed, *U _{s}* (see Fig. S2 in the supplementary material). Hence, the capillary pressure, $\u2248\sigma (\theta 2\u2212\theta s2)$, will increase and further drive the contact line faster.

^{9}Therefore, increasing the contact angle with the plunging speed causes the contact line to spread faster over the sphere. Additionally, it is worth mentioning that the linear relation can be shown in the impact-inertial balance between the inertia ($\u223c\rho lUc2$) on the contact line and the impact pressure ($\u223c\rho lUs2$) of a sphere as well.

^{24}Similarly, in the case of a drop impact, the contact line velocity is observed to be the impact speed in high inertia regimes as the drop-impact pressure balances with inertia in the drop.

^{24}

In this study, we investigated the dynamics of the contact line on a sphere as it penetrates through the liquid/air interface. From experiments using optical and x-ray light sources, three distinct spreading stages were identified. In the very beginning, just after the meniscus rises, the capillary-inertial velocity is observed, which is independent of the sphere-velocity. Then, the contact line speed decreases and reaches a terminal velocity. This terminal velocity is set by the plunging speed of the sphere. When the Ohnesorge number is higher than 0.02, the contact line velocity and angle follow the capillary-viscous model as a balance between capillary and viscous stresses. However, with Oh < 0.02, the contact line velocity gets smaller than the velocity predicted in the capillary-viscous model as inertia plays a role.

See supplementary material for reduction in data scattering from the capillary-viscous model by choosing different cut-off Weber numbers (Fig. S1) and contact-angle adjustment by a sphere-sinking speed (Fig. S2). Tables S1 and S2 list physical and non-dimensional parameters used in the present experiments. The subplots of Fig. 4 are provided for different liquids and solid materials (Figs. S3 and S4) and Bond numbers (Fig. S5).

The authors would like to thank Dr. S. Gart, Dr. P. Zhang, Dr. J. Boreyko, B. Chang, S. Poulain, Z. Zhang, and A. Deriy for their help in the present work. This work was supported by the National Science Foundation (CBET-1336038 and CBET-1604424). This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by the Argonne National Laboratory under Contract No. DE-AC02-06CH11357.