ALBERT

Description: 〈div data-abstract-type="normal"〉〈p〉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 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline1.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 and slip length 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline2.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉, the apparent dynamic contact angle 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline3.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 can change from Tanner’s law 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline4.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 for 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline5.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 to the strong-slip law 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline6.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 for 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline7.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉, where 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline8.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 is the capillary number and 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline9.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 is the macroscopic length scale. Such a no-slip-to-slip transition occurs at the critical capillary number 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline10.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉, accompanied by the switch of the ‘foot’ of size 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline11.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 from the inner scale to the outer scale with respect to 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline12.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉. A more generalized dynamic contact angle relationship is also derived, capable of unifying Tanner’s law and the strong-slip law under 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline13.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉. 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, 〈span〉J. Colloid Interface Sci〈/span〉., vol. 122, 1988, pp. 60–72; Albrecht 〈span〉et al.〈/span〉, 〈span〉Phys. Rev. Lett.〈/span〉, 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 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline14.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 followed by a mesoscopic ‘foot’ of width 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline15.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 ahead of the macroscopic wedge, where 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline16.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 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 〈span〉et al.〈/span〉 (〈span〉Phys. Rev. Lett.〈/span〉, vol. 91, 2003, 196104) and Ueno 〈span〉et al.〈/span〉 (〈span〉Trans. ASME J. Heat Transfer〈/span〉, vol. 134, 2012, 051008). The advancement of the microscopic contact line is still led by the film whose length can grow as the 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline17.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 power of time due to 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190619140754267-0248:S0022112019003525:S0022112019003525_inline18.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉, as supported by the experiments of Ueno 〈span〉et al.〈/span〉 and Mate (〈span〉Langmuir〈/span〉, 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.〈/p〉〈/div〉

Description: 〈div data-abstract-type="normal"〉〈p〉When there exists slip on the surface of a solid body moving in an unsteady manner, the extent of slip is not fixed but constantly changes with the time-varying Stokes boundary layer thickness 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190312090638125-0973:S0022112019000570:S0022112019000570_inline1.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 in competition with the slip length 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190312090638125-0973:S0022112019000570:S0022112019000570_inline2.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉. Here we revisit the unsteady motion of a slippery spherical particle to elucidate this dynamic slip situation. We find that even if the amount of slip is minuscule, it can dramatically change the characteristics of the history force, markedly different from those due to non-spherical and fluid particles (Lawrence & Weinbaum, 〈span〉J. Fluid Mech.〈/span〉, vol. 171, 1986, pp. 209–218; Yang & Leal, 〈span〉Phys. Fluids〈/span〉 A, vol. 3, 1991, pp. 1822–1824). For an oscillatory translation of such a particle of radius 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190312090638125-0973:S0022112019000570:S0022112019000570_inline3.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉, two distinctive features are identified in the frequency response of the viscous drag: (i) the high-frequency constant force plateau of 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190312090638125-0973:S0022112019000570:S0022112019000570_inline4.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 much greater than the steady drag due to a constant shear stress caused by 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190312090638125-0973:S0022112019000570:S0022112019000570_inline5.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 much thinner than 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190312090638125-0973:S0022112019000570:S0022112019000570_inline6.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 and (ii) the persistence of the plateau while lowering the frequency until the slip–stick transition point 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190312090638125-0973:S0022112019000570:S0022112019000570_inline7.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉, beyond which 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190312090638125-0973:S0022112019000570:S0022112019000570_inline8.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 becomes thicker and the usual Basset decay reappears. Similar features can also be observed in the short-term force response for the particle subject to a sudden movement, as well as in the behaviour of the torque when it undergoes rotary oscillations. In addition, for both translational and rotary oscillations, slip can further introduce a phase jump from the no-slip value to zero in the high-frequency limit. As these features and the associated slip–stick transitions become more evident as 〈span〉〈span〉〈img data-mimesubtype="gif" data-type="simple" src="http://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20190312090638125-0973:S0022112019000570:S0022112019000570_inline9.gif"〉 〈span data-mathjax-type="texmath"〉 〈/span〉 〈/span〉〈/span〉 becomes smaller and are exclusive to the situation where surface slip is present, they might have potential uses for extracting the slip length of a colloidal particle from experiments.〈/p〉〈/div〉