Moving contact lines and dynamic contact angles: a ‘litmus test’ for mathematical models, accomplishments and new challenges

Abstract

After a brief overview of the ‘moving contact-line problem’ as it emerged and evolved as a research topic, a ‘litmus test’ allowing one to assess adequacy of the mathematical models proposed as solutions to the problem is described. Its essence is in comparing the contact angle, an element inherent in every model, with what follows from a qualitative analysis of some simple flows. It is shown that, contrary to a widely held view, the dynamic contact angle is not a function of the contact-line speed as for different spontaneous spreading flows one has different paths in the contact angle-versus-speed plane. In particular, the dynamic contact angle can decrease as the contact-line speed increases. This completely undermines the search for the ‘right’ velocity-dependence of the dynamic contact angle, actual or apparent, as a direction of research. With a reference to an earlier publication, it is shown that, to date, the only mathematical model passing the ‘litmus test’ is the model of dynamic wetting as an interface formation process. The model, which was originated back in 1993, inscribes dynamic wetting into the general physical context as a particular case in a wide class of flows, which also includes coalescence, capillary breakup, free-surface cusping and some other flows, all sharing the same underlying physics. New challenges in the field of dynamic wetting are discussed.

Effective boundary conditions for dynamic contact angle hysteresis on chemically inhomogeneous surfaces

“…However, these conditions might fail to predict the experimental observations when the flow field is very complex near the contact line (Blake, Bracke & Shikhmurzaev 1999;Clarke & Stattersfield 2006;Davis & Davis 2013;Mohammad Karim, Davis & Kavehpour 2016). This fact has been highlighted in a recent work (Shikhmurzaev 2020). There, the author presents some examples as a 'litmus test' for the mathematical models for moving contact lines and concludes that the standard models like the Cox's boundary condition cannot explain the contact line motion in all situations.…”

Revisiting Shikhmurzaev’s Approach to the Contact Line Problem

“…equation (3.25) 2 ). This above model of meniscus formation as a general phenomena (not the thin film approximation) has been mentioned in [29,Fig. 6].…”

The Cox–Voinov law for traveling waves in the partial wetting regime*

“…If on the other hand γ gs γ ls + γ gl , then θ = 0 (zero contact angle), a global equilibrium is not attained, and the thin film eventually covers the entire solid (complete-wetting regime). While microscopically Young's law (1.2) applies, the apparent macroscopic contact angle is dynamic and in general depends on the flow (for instance through the velocity at the contact line, cf [53] and references therein). The difference is schematically visualized in figure 3.…”