In a 1997 note, E. Raphael and P.-G. de Gennes proposed controlling the wettability of a periodically grooved solid with a magnetizable liquid (a ferrofluid): an applied magnetic field changes the equilibrium impregnation of the grooves and hence the effective wetting angle θH\theta_{H}, and can drive the invasion of the texture by the liquid. Their theory is scaling-level and rests on two acknowledged approximations: a variational ansatz (a polynomial free-surface profile, eq. (5) of the note) with an essentially constant-slope surface, valid only when the film aspect ratio h/2bh/2b is not too small (hh = film thickness, 2b2b = groove spacing scale), and a coarse-grained front dynamics (eq. (18)) that assumes an edge-trapped film of constant thickness e0e_{0} far behind the advancing front and an equilibrium contact angle θe\theta_{e} at it, leaving the microscopic contact-line region, its dissipation, and several numerical constants unspecified. The open problem is to complete this theory into a quantitative description validated against external evidence: derive the accurate equilibrium film profiles and θH(H)\theta_{H}(H) in the thin-film regime h≪2bh\ll 2b where the constant-slope ansatz fails; construct the microscale analysis and dissipation closure for the field-driven impregnation front in the region where the local film thickness tends to zero and edge-trapping breaks down; determine the omitted geometric constants and numerical prefactors; and test the resulting static and dynamic scaling laws against external evidence. No work addressing this specific combination is known in the literature to date.

Background

Wettability is the tendency of a solid to be wet by a liquid, quantified by the contact angle θ\theta at which the liquid-vapor interface meets the solid; Young's law γcos⁡θ=γSV−γSL\gamma\cos\theta=\gamma_{SV}-\gamma_{SL} sets the equilibrium angle from the interfacial tensions γ\gamma (liquid-vapor), γSV\gamma_{SV} and γSL\gamma_{SL} (solid-vapor, solid-liquid). On a textured surface, for example a solid carved with a periodic array of grooves of spacing 2b2b, the liquid can either sit on top of the texture (a Cassie state) or fill the grooves; the latter is called impregnation or hemi-wicking, and it changes the apparent, coarse-grained contact angle of a drop or film because part of the solid is replaced by a liquid-vapor interface. A ferrofluid is a colloidal suspension of single-domain magnetic nanoparticles that behaves as a uniformly magnetizable liquid; its magnetic energy density, of order μ0χH2\mu_{0}\chi H^{2} for susceptibility χ\chi and field HH, can compete with the capillary energy γ\gamma per unit area, so applied fields deform ferrofluid interfaces and modify their effective wetting (magnetowetting), a subject systematized in ferrohydrodynamics. Field-controlled wetting of simple geometries was demonstrated early: a ferrofluid in contact with a wire undergoes a field-driven wetting transition (Bacri et al., 1988), and the shape of a ferrofluid meniscus in a fringing magnetic field was computed by ferrohydrostatics, the static equilibrium theory of magnetized liquids (Boudouvis et al., 1988). The framework of capillarity, contact-angle hysteresis, wetting of textured surfaces, and contact-line dynamics used throughout is standard and is collected in the monograph by de Gennes, Brochard-Wyart and Quere (2004).

Against this background, Raphael and de Gennes (1997) considered the magnetic control of the wettability of a grooved solid by such a magnetizable liquid. In their picture the liquid partially impregnates the grooves; a magnetic field biases the impregnation equilibrium and produces an effective, field-dependent wetting angle θH\theta_{H} for the film covering the texture. Their statics used a variational calculation of the film's free-surface profile with a polynomial ansatz (eq. (5) of the note), a description whose constant-slope surface profile is plausible only when the film aspect ratio h/2bh/2b is not too small. Their dynamics described the field-driven advance of the liquid front invading the grooves. Two ingredients were essential there: edge pinning, i.e. the trapping of the contact line on the sharp edges of the grooves (a standard consequence of the Gibbs pinning condition at a solid edge), which lets the invaded film far behind the front settle to a constant thickness e0e_{0}; and the assumption that the contact angle at the front keeps its equilibrium value θe\theta_{e}. Averaging over many grooves then yields a coarse-grained partial differential equation (eq. (18)) for the front. The authors themselves flagged what the theory does not do. First, in the thin-film limit h≪2bh\ll 2b the free surface departs from the constant-slope picture and the ansatz-based θH\theta_{H} must be replaced by a more accurate calculation. Second, near the advancing front the local film thickness tends to zero, the edge-trapping hypothesis fails, and the microscopic contact-line behavior and its viscous dissipation (which in wetting dynamics are concentrated in the small-scale wedge region and control the kinetic coefficient of the front) are not analyzed beyond the coarse-grained PDE. Third, some parameters of the hydrodynamic description, notably a geometric curvature length d0d_{0} and several numerical prefactors left as unspecified constants, were not determined, and no experimental test of the scaling predictions was made. The question below asks for the completion of this program: it is one coherent objective, since the same balance of capillary, magnetic, and viscous forces in the same groove geometry determines the static profile, the microscopic front dynamics, and the constants that make the theory predictive.

Problem Statement

For a magnetizable liquid (ferrofluid) in contact with a periodically grooved solid surface, whose film of thickness hh partially impregnates grooves of spacing scale 2b2b under an applied magnetic field HH, complete the quantitative theory of magnetic control of the wettability initiated by Raphael and de Gennes (1997), to the level of a predictive, experimentally validated description. This comprises three linked strands that together close the acknowledged gaps of the 1997 note:

  1. Statics in the thin-film regime. Derive the accurate equilibrium free-surface profile of the liquid film covering the grooved surface, and the resulting effective, field-dependent wetting angle θH\theta_{H}, in the limit h≪2bh\ll 2b where the constant-slope polynomial variational ansatz (eq. (5) of the note) ceases to be valid. The derivation should make explicit the balance of capillary and magnetic energies — including the magnetization behavior of the liquid, e.g. demagnetizing effects (the reduction of the internal field inside the film caused by its own magnetization) and, where relevant, nonlinear susceptibility or saturation of the magnetization — and should reduce to the 1997 result and to the zero-field capillary results in the appropriate limits.

  2. Microscopic front dynamics. Construct the microscale analysis of the advancing front of the liquid invading the grooves, in the region where the local film thickness tends to zero and the trapping of the liquid on the groove edges fails, including the microscopic contact-line behavior and the viscous dissipation concentrated there, so that the coarse-grained front equation (eq. (18) of the note) acquires a controlled microscopic closure rather than undetermined phenomenological coefficients. The relevant microscopic scale set by the groove-edge geometry (the curvature length denoted d0d_{0} in the note) should emerge from the analysis.

  3. Constants and validation. Determine the geometric constants and numerical prefactors left unspecified in the 1997 hydrodynamic description, and test the resulting static and dynamic scaling predictions for a characterized grooved surface and a characterized ferrofluid (equilibrium film profiles and θH(H)\theta_{H}(H), and front position versus time under controlled fields) against evidence external to the derivation: either reanalysis of measured profiles and front-position data, or an independent higher-fidelity computation of the same geometry and magnetization law. A self-consistency check of the derivation's own asymptotics does not discharge this strand.

The common objective uniting the three strands is a single quantitative theory — the same groove geometry and the same balance of capillary, magnetic, and viscous forces determine the static profile, the microscopic front dynamics, and the constants — of magnetically controlled wettability and impregnation of a grooved surface by a magnetizable liquid. An answer that solves, say, only the statics on an unrelated geometry, or that replaces the stated objective by a proxy (e.g. droplet shapes on flat surfaces), does not answer this problem.

References

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  10. B. Bolteau, "La magnétisante histoire de la goutte fakir ou étude des propriétés de mouillage de surfaces superhydrophobes à géométrie magnétiquement modulable" (The magnetizing story of the fakir drop: wetting properties of magnetically actuated superhydrophobic surfaces), PhD thesis, Sorbonne Université (2018). NNT: 2018SORUS062. DOI: 10.70675/48921578z41f7z496dzbdfaz3ba03f241c54, https://theses.hal.science/tel-02163924
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