Water Film Drainage between a Very Viscous Oil Drop and a Mica Surface

We investigate thin film drainage between a viscous oil drop and a mica surface, clearly illustrating the competing effects of Laplace pressure and viscous normal stress (${\tau }_v$) in the drop. ${\tau }_v$ dominates the initial stage of drainage, leading to dimple formation ($h_d$) at a smaller critical thickness with an increase in the drop viscosity (the dimple is the inversion of curvature of the drop in the film region). Surface forces and interfacial tension control the last stage of film drainage. A scaling analysis shows that $h_d$ is a function of the drop size $R$ and the capillary numbers of the film (${\mathrm{Ca}}_f$) and drop (${\mathrm{Ca}}_d$), which we estimate by $h_d=0.5R\sqrt{{\mathrm{Ca}}_f/(1+2{\mathrm{Ca}}_d)}$. This equation clearly indicates that the drop viscosity needs to be considered when ${\mathrm{Ca}}_d>0.1$. These results have implications for industrial systems where very viscous liquids are involved, for example, in 3D printing and heavy oil extraction process.

Figure 1

(a) Schematic of the dynamic force apparatus. An oil drop with radius $R=1.05\pm 0.01\mathrm{mm}$ was generated at the end of a capillary. The drop was driven toward the hydrophilic mica surface by a motor. (b) Thin film region corresponding to the red square in (a). (c) A snapshot of the interference fringes (green channel) obtained between an oil drop and a mica surface (${\mu }_o=25.8\mathrm{Pa}\mathrm{s}$, $V=1.06\mathrm{mm}/\mathrm{s}$) in 0.1 mM SDS solution. (d) Axisymmetric film thickness profile obtained from (c).

Figure 2

(a) Comparison between the experimental results (points) and the theoretical model (lines) for the film evolution of an oil drop of 0.001 $\mathrm{Pa}\mathrm{s}$ viscosity interacting with a mica surface in 0.1 mM SDS aqueous solution at the approach velocity of $1.06\mathrm{mm}/\mathrm{s}$ (${\mathrm{Ca}}_d={10}^{-5}$). The measured times of the profiles from top to bottom are $-0.015$, 0.018, 0.085, 0.218, 0.55, 2.12, 9.07, 31.8, 66.4, 153.5, 211.9, and 331.7 s. The dashed line indicates when the oil drop stopped moving. (b) Film drainage process using an oil drop of 37.0 $\mathrm{Pa}\mathrm{s}$ viscosity in 0.1 mM SDS aqueous solution at the approach velocity of $1.06\mathrm{mm}/\mathrm{s}$ (${\mathrm{Ca}}_d=0.8$). The measured times of the profiles from top to bottom are $-0.001$, 0, 0.067, 0.618, 19.7, 97.1, 144.5, 211.0, 318.1, and 391.6 s. Drop stopped moving at 0.26 s.

Figure 3

(a) Film thickness at the center $h(0,t)$ as a function of time for oil drops of different viscosity in 0.1 mM SDS solution at an approach velocity of $1.06\mathrm{mm}/\mathrm{s}$. (b) Selected drop shape profiles of an oil drop of 37.0 $\mathrm{Pa}\mathrm{s}$ viscosity from Fig. 2 in 0.1 mM SDS aqueous solution at the approach velocity of $1.06\mathrm{mm}/\mathrm{s}$ at times 0.067, 0.618, and 19.7 s.

Figure 4

Height of initial dimple formation for oil drops or bubbles interacting with a hydrophilic solid surface in water, as a function of ${\mathrm{Ca}}_d$ as compared with the theoretical curve of Eq. (8) (line). The five different cases are (a) $V=1.06\mathrm{mm}/\mathrm{s}$ in 0.1 mM SDS solution, ${\mu }_o=0.001–111.5\mathrm{Pa}\mathrm{s}$; (b) $V=0.1\mathrm{mm}/\mathrm{s}$ in 0.1 mM SDS solution, ${\mu }_o=0.001–111.5\mathrm{Pa}\mathrm{s}$; (c) $V=1.06\mathrm{mm}/\mathrm{s}$ in 1 mM SDS solution, ${\mu }_o=0.001–111.5\mathrm{Pa}\mathrm{s}$; (d) ${\mu }_o=111.5\mathrm{Pa}\mathrm{s}$ in 0.1 mM SDS solution, $V=0.1–10\mathrm{mm}/\mathrm{s}$; (e) bubble interacting with silica in water. The data of the dimple height for bubble interacting with hydrophilic silica surface in water (e) is from Ref. [8]. The result of the theoretical calculation fit well with the results from corresponding experiments. The transition occurred at ${\mathrm{Ca}}_d$ around 0.1. Inset: the height of initial dimple formation as a function of oil viscosity (same legend).