Slipping moving contact lines: critical roles of de Gennes’s ‘foot’ in dynamic wetting

Abstract

In the context of dynamic wetting, wall slip is often treated as a microscopic effect for removing viscous stress singularity at a moving contact line. In most drop spreading experiments, however, a considerable amount of slip may occur due to the use of polymer liquids such as silicone oils, which may cause significant deviations from the classical Tanner–de Gennes theory. Here we show that many classical results for complete wetting fluids may no longer hold due to wall slip, depending crucially on the extent of de Gennes’s slipping ‘foot’ to the relevant length scales at both the macroscopic and microscopic levels. At the macroscopic level, we find that for given liquid height $h$ and slip length $\unicode[STIX]{x1D706}$ , the apparent dynamic contact angle $\unicode[STIX]{x1D703}_{d}$ can change from Tanner’s law $\unicode[STIX]{x1D703}_{d}\sim Ca^{1/3}$ for $h\gg \unicode[STIX]{x1D706}$ to the strong-slip law $\unicode[STIX]{x1D703}_{d}\sim Ca^{1/2}\,(L/\unicode[STIX]{x1D706})^{1/2}$ for $h\ll \unicode[STIX]{x1D706}$ , where $Ca$ is the capillary number and $L$ is the macroscopic length scale. Such a no-slip-to-slip transition occurs at the critical capillary number $Ca^{\ast }\sim (\unicode[STIX]{x1D706}/L)^{3}$ , accompanied by the switch of the ‘foot’ of size $\ell _{F}\sim \unicode[STIX]{x1D706}Ca^{-1/3}$ from the inner scale to the outer scale with respect to $L$ . A more generalized dynamic contact angle relationship is also derived, capable of unifying Tanner’s law and the strong-slip law under $\unicode[STIX]{x1D706}\ll L/\unicode[STIX]{x1D703}_{d}$ . We not only confirm the two distinct wetting laws using many-body dissipative particle dynamics simulations, but also provide a rational account for anomalous departures from Tanner’s law seen in experiments (Chen, J. Colloid Interface Sci., vol. 122, 1988, pp. 60–72; Albrecht et al., Phys. Rev. Lett., vol. 68, 1992, pp. 3192–3195). We also show that even for a common spreading drop with small macroscopic slip, slip effects can still be microscopically strong enough to change the microstructure of the contact line. The structure is identified to consist of a strongly slipping precursor film of length $\ell \sim (a\unicode[STIX]{x1D706})^{1/2}Ca^{-1/2}$ followed by a mesoscopic ‘foot’ of width $\ell _{F}\sim \unicode[STIX]{x1D706}Ca^{-1/3}$ ahead of the macroscopic wedge, where $a$ is the molecular length. It thus turns out that it is the ‘foot’, rather than the film, contributing to the microscopic length in Tanner’s law, in accordance with the experimental data reported by Kavehpour et al. (Phys. Rev. Lett., vol. 91, 2003, 196104) and Ueno et al. (Trans. ASME J. Heat Transfer, vol. 134, 2012, 051008). The advancement of the microscopic contact line is still led by the film whose length can grow as the $1/3$ power of time due to $\ell$ , as supported by the experiments of Ueno et al. and Mate (Langmuir, vol. 28, 2012, pp. 16821–16827). The present work demonstrates that the behaviour of a moving contact line can be strongly influenced by wall slip. Such slip-mediated dynamic wetting might also provide an alternative means for probing slippery surfaces.

Slip-enhanced Rayleigh–Plateau instability of a liquid film on a fibre

“…Based on the approach proposed by Liao et al (2013) and Wei, Tsao & Chu (2019), we have a physical explanation for why large slip causes the failure of lubrication models, i.e. the radial derivative of the axial velocity is no longer much larger than its axial derivative, violating the lubrication approximation |∂ 2 u/∂x 2 |/|(1/r)(∂/∂r)(r(∂u/∂r))| 1 (see more details in Appendix A).…”

“…Note that the lubrication equation (A21) is not available for a large-slip case because ls ∼ ε λ, derived from (A8a,b). Moreover, ls was found to be not arbitrarily large, but has to be under a certain constraint to make the lubrication approximation hold (Liao et al 2013;Wei et al 2019). To derive such a constraint for (A21), we start from (A3), where the radial derivative of the axial velocity is found to be much larger than its axial derivative:…”

“…With an approach similar to that employed by Liao et al (2013) and Wei et al (2019), the radial derivative is scaled as…”

“…2013; Wei et al. 2019). To derive such a constraint for (A21), we start from (A3), where the radial derivative of the axial velocity is found to be much larger than its axial derivative: Expanding the radial derivative yields With an approach similar to that employed by Liao et al.…”

“…(2013) and Wei et al. (2019), the radial derivative is scaled as where is the dimensional characteristic velocity, and . The axial one is Substituting (A26)–(A28) into (A25) gives the final constraint, whose dimensionless form is …”