Thin-Film Equations with Singular Potentials: An Alternative Solution to the Contact-Line Paradox

In the regime of lubrication approximation, we look at spreading phenomena under the action of singular potentials of the form P(h)≈h1-m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(h)\approx h^{1-m}$$\end{document} as h→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\rightarrow 0^+$$\end{document} with m>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>1$$\end{document}, modeling repulsion between the liquid–gas interface and the substrate. We assume zero slippage at the contact line. Based on formal analysis arguments, we report that for any m>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>1$$\end{document} and any value of the speed (both positive and negative) there exists a three-parameter, hence generic, family of fronts (i.e., traveling-wave solutions with a contact line). A two-parameter family of advancing “linear-log” fronts also exists, having a logarithmically corrected linear behavior in the liquid bulk. All these fronts have finite rate of dissipation, indicating that singular potentials stand as an alternative solution to the contact-line paradox. In agreement with steady states, fronts have microscopic contact angle equal to π/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi /2$$\end{document} for all m>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>1$$\end{document} and finite energy for all m<3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m<3$$\end{document}. We also propose a selection criterion for the fronts, based on thermodynamically consistent contact-line conditions modeling friction at the contact line. So as contact-angle conditions do in the case of slippage models, this criterion selects a unique (up to translation) linear-log front for each positive speed. Numerical evidence suggests that, fixed the speed and the frictional coefficient, its shape depends on the spreading coefficient, with steeper fronts in partial wetting and a more prominent precursor region in dry complete wetting.

It is about half a century since the no-slip paradox was discovered by Huh and Scriven [1971] and Dussan V. and Davis [1974].The paradox may be summarized as follows: take a liquid droplet which is sliding over a solid dry substrate, as modeled by Stokes equations; if the no-slip condition were adopted, that is, if a null liquid's horizontal velocity were prescribed at the liquid-solid interface, then an infinite force would be required to move the contact line, i.e. the triple junction where solid, liquid and gas meet.In the words of Huh and Scriven [1971], "not even Herakles could sink a solid".The paradox may be discussed already in the framework of lubrication approximation: this is an asymptotic limit of the full Navier-Stokes system under a suitable scaling, which in words requires a small ratio of vertical vs horizontal length-scale, a relatively small velocity, and a relatively large surface tension.It is a simplified model which however retains the essential physics at such scales: a dissipative evolution driven by surface tension and limited by both viscous and interfacial friction (see for example (1.7), (3.4), and (3.13) below).In fact, generic contact lines, at leading order around one of their points, are locally straight: therefore the essential features of the contact-line paradox are already captured by a one-dimensional setting, which we therefore adopt in the sequel.
1.1.The model.In lubrication theory, the evolution of a thin liquid film, or a droplet, of viscous incompressible liquid over a horizontal solid substrate is described by the thinfilm equation [Greenspan, 1978, Hocking, 1983, Oron et al., 1997, Giacomelli and Otto, 2003, Knüpfer and Masmoudi, 2015], which in its basic one-dimensional form reads as x on {h > 0}. (1.1) Here the solid substrate corresponds to the x-axis, t is time, h(t, x) is the liquid's height over the solid, µ is the liquid's viscosity, and γ is the liquid-gas surface tension.The mobility function m depends on the condition at the liquid-solid interface: when the no-slip condition is assumed, then m(h) = 1 3 h 3 . (1. 2) The potential Q usually combines the effects of intermolecular, surface, and gravitational forces [ de Gennes, 1985]; here we shall ignore the latter ones for simplicity: with G ≡ 0 and S ∈ R in this manuscript.(1. 3) The constant S (assumed to be relatively small in lubrication theory, cf.Remark 1.2) is the non-dimensional spreading coefficient: where γ SL , and γ SG are the solid-liquid and solid-gas tensions, respectively.There is, however, a caveat to be made at this point.In thermodynamic equilibrium of the solid with the surrounding vapor phase (the so-called "moist" case, which concerns for instance a surface which has been pre-exposed to vapor), γ SG is usually denoted by γ SV , and its value can never exceed γ SL + γ.Indeed, otherwise the free energy of a solid/vapor interface could be lowered by inserting a liquid film in between: the equilibrium solid/vapor interface would then comprise such film, leading to γ SV = γ SL + γ.Therefore, S ≤ 0 in the "moist" case.On the other hand, when the solid and the gaseous phases are not in thermodynamical equilibrium (the so-called "dry" case), there is no constraint on the sign of S.
The function P is an intermolecular potential.Generally speaking, P is singular as h → 0 + and decays to zero as h → +∞, with Π = −P usually referred to as the disjoining pressure.We consider the case in which P is short-range repulsive, in the sense that it penalizes short distances between the liquid-gas interface and the solid:1 P (h) ∼ A m−1 h 1−m as h → 0 + , A > 0, m > 1, P (0) = P (+∞) = 0. (1.4) The standard choice for P yields m = 3: , where A > 0 is the Hamaker constant, (1.5) which corresponds to an integration of Lifshitz-van der Waals interactions between molecules [Israelachvili, 2011, Craster andMatar, 2009] and which we shall hereafter refer to as van der Waals potentials.There are, however, reasons to examine different values of m.The first one is that the form of the disjoining pressure is highly dependent on the nature of the dominant intermolecular force (molecular, electrostatic, structural) and on the scales under consideration: for instance, the electrostatic and structural contributions for water on glass or silica surfaces yield m = 1 or m = 2, depending on thickness [Pashley, 1980, Teletzke et al., 1988]; we refer to the lucid discussion in Dallaston et al. [2018].A second reason will be introduced in §1.3.
1.2.The contact-line paradox.In order to introduce the contact-line paradox, it is convenient to describe the basic energetic structure of (1.1).The free energy of the system is given by In lubrication theory, the term γ(1+ 1 2 h 2 x ) is the leading-order approximation of the liquidgas surface energy density γ 1 + h 2 x .The summand 1 is incorporated in the potential Q. Smooth, positive and, say, periodic solutions to (1.1) (e.g.modelling a liquid film) satisfy the energy balance V 2 dx rate of bulk dissipation . (1.7) When non-negative solutions are considered (e.g.modelling a droplet), then contributions at the contact line appear (cf.e.g.(3.10) below), but the rate of bulk dissipation remains the same: it encodes both viscous friction within the liquid and, through m, interfacial friction at the liquid-solid interface.
In the framework of (1.1), the contact-line paradox manifests itself as follows.Assume the no-slip condition, i.e. (1.2), and consider a travelling wave solutions to (1.1): where V = 0 is a constant velocity (here we have assumed w.l.o.g. that the contact line is initially located at x = 0).If P ≡ 0, then advancing (V > 0) travelling wave solutions to (1.1) of the form (1.8) do not exist, whereas receding ones (V < 0) have a non-integrable rate of dissipation density (see (2.31) below).Therefore, for m(h) = 1 3 h 3 and P ≡ 0, travelling waves with finite dissipation do not exist at all: this is the manifestation of the contact-line paradox in lubrication theory.
Since the contact-line paradox was discovered, quite a few enrichments of the basic model have being put forward to relieve it: we refer to the reviews by Oron et al. [1997], de Gennes [1985], Bonn et al. [2009], and Snoeijer and Andreotti [2013].The standard method, which was first investigated by Huh and Mason [1977], Hocking [1976Hocking [ , 1977]], and Greenspan [1978], is to allow for fluid slip over the solid (at least in a neighborhood of the contact line), which amounts to prescribing a relation between horizontal velocity and shear stress at the liquid-solid interface.In lubrication theory, these relations modify the mobility function to m(h) = 1 3 (h 3 + λ 3−n h n ) (n = 2 for the classical Navier [1823] slip condition); λ is a length-scale whose inverse is proportional to liquid-solid friction.A second one, introduced to our knowledge by Weidner and Schwartz [1994], is to assume a shear-thinning rheology, with a vanishing liquid's viscosity as the shear stress blows up: this introduces a nonlinear dependence of V on h xxx (see e.g.Flitton and King [2004], King [2001], Ansini andGiacomelli [2002, 2004]).More recently, it has been observed by Rednikov andColinet [2013, 2019] and Janeček et al. [2013Janeček et al. [ , 2015] ] that the Kelvin effect, i.e., a curvature-induced variation of saturation conditions, may also be employed to resolve contact-line paradox: in this case, (1.1) is complemented with a singular term in non-divergence form.
Another way to resolve the paradox was first discussed by Starov [1983], de Gennes [1984], and Hervet and de Gennes [1984]: it consists in taking the effect of intermolecular potentials P into account.The goal of this note is to revisit this phenomenon in a systematic way for generic potentials.To this aim, it is convenient to review the statics first.
1.3.Statics.Consider absolute minimizers h min of E, as given by (1.6), under the constraint of given mass M .
When P ≡ 0 and S < 0, h min is an arc of parabola characterized by its mass M and |S|; in particular, its slope at the contact line ∂{h > 0} is determined by |S|: , θ S := arctan 2|S|, S < 0.
When P ≡ 0 and S ≥ 0, h min instead does not exist, and minimizing sequences converge to an unbounded film with zero thickness.Therefore it is common to define the static (or equilibrium) microscopic contact angle θ S , and to name the two regimes, as follows: (1.9) Let us now take P into account.Consider steady states h ss with connected positivity set, that is, solutions to the Euler-Lagrange equation where Λ ∈ R is a Lagrange multiplier coming from the mass constraint.The properties of h ss have been formally discussed by Joanny and de Gennes [1984], de Gennes [1985], and Leger and Joanny [1992].Their macroscopic shape (here encoded by looking at the regime M 1) may be droplet-like or pancake-like.When R(h) = Q(h)/h has a unique absolute minimum point e * , then e * is characterized by S = P (e * ) − e * P (e * ) (1.11) and h ss is pancake-shaped: (1.12) Since S > 0 implies e * < +∞, this configuration is generic in dry complete wetting.Thus, it is the wetting coefficient which drives the system towards a "pancake" equilibrium.If on the other hand R has no absolute minimum, then necessarily S ≤ 0; steady states are droplet-shaped if S < 0, (1.13) where θ mac denotes the macroscopic contact angle (see Fig. 1).When S = 0 and e * = +∞ the shape is still droplet-like, but constants depend on the large-h behavior of P .Finally, a simple asymptotic expansion of (1.10) using (1.4) shows that implying that the microscopic contact angle equals π/2 for any m > 1 (Fig. 1).If the model is assumed to hold down to h = 0, the above characterization suffers from a limitation if m ≥ 3. Indeed, it follows from (1.14) that both summands in the energy, h 2 x and Q(h), are not integrable for m ≥ 3. Therefore, steady states have unbounded energy if m ≥ 3: (1.15) The dual of this phenomenon is the following: if m ≥ 3 and h has finite energy and positive mass, then h can not tend to zero, either at any point or at infinity (cf.Lemma 1 in the Appendix).
(1.16)This yields the variational counterpart of (1.15): mass-constrained minimizers of E in H 1 (R) do not exist if m ≥ 3. (1.17) On the other hand, if the singularity is milder, that is if 1 < m < 3, it was recently observed in Durastanti and Giacomelli [2022] that compactly supported minimizers h min do exist, and coincide with steady states with connected positivity set.In particular, h min satisfies (1.12) and (1.13) with e * defined by (1.11).In summary: Steady states with connected positivity set exist for all m > 1 and satisfy (1.12), (1.13) and (1.14); however, their energy is finite and minimal if and only if m < 3.
(1.18) When m ≥ 3, as for the van der Waals potential P 0 , the limitation in (1.15)-(1.17) is usually handled by arguing that, at scales below a few molecules' radii (say, about ten Ångström for water), a continuum description of molecular interaction through P 0 may not be valid any more, since it is based on integration of binary molecular interactions.Hence the validity of a continuum description such as (1.1) is taken only up to a molecular threshold length-scale ( 2 = a 2 = A/6πγ in de Gennes [1985], Gennes et al. [1990], and Leger and Joanny [1992]).
On the other hand, (1.15)-(1.17)can not be ignored when one assumes m ≥ 3 and seeks for a continuum model which consistently describes the liquid's profile all the way down to h = 0, capturing "pancake" shapes in dry complete wetting (S > 0).Unfortunately, simple fixes do not work.Indeed, either setting a virtual "zero height" at by the translation ĥ = h − , or introducing a naive cut-off of the potential, such as P (h) = min{P 0 (h), P 0 ( )}, make (1.1) non-singular; hence the final spreading equilibrium will not be a pancake either (mass-constrained minimizing sequences tend to zero: in more suggestive terms, the equilibrium is an unbounded layer of zero thickness).A much more ingenious fix dates back to the work of Bertozzi and Pugh [1994] and consists in introducing a less singular "molecular cut-off", such as where P 0 is as in (1.5) and is a molecular-sized threshold length-scale.Based on (1.18), we expect that (1.19) may provide an equally effective description of droplets' profiles in the framework of van der Waals potentials, without the disadvantage of an infinite energy.
The goal of this manuscript is to explore, in the dynamical framework, a parallel between m ≥ 3 and m < 3 analogous to the one in (1.18).The preliminary matter in dynamical studies is of course that of travelling wave solutions, which will be discussed in §2.In §3 we will identify a class of thermodynamically consistent contact-line conditions modelling contact-line friction, in the spirit of Ren and E [2007], Ren et al. [2010], Ren and E [2011].Finally, in §4 we will draw our conclusions and present quite a few open questions.All of our observations will be supported by numerical examples.
Remark 1.1.An alternative approach to the contact-line paradox, which is quite common in the applied math community, is to take advantage of (1.16): when their validity down to h = 0 + is assumed, potentials which are sufficiently repulsive at h = 0 + and attractive at ∞, e.g. of the form yield periodic steady states which consist of arrays of droplets over a microscopic film of thickness O(h * ) fully covering the substrate.Thus (1.20) circumvents, rather than solving, the paradox.However, (1.20) has proved to be particularly fruitful in numerical simulations and asymptotic studies, mainly in relation to dewetting phenomena: (in)stability of the flat film, bifurcation, concentration, and asymptotic scaling laws with respect to the potential's parameters,; see Bertozzi et al. [2001], Laugesen and Pugh [2002], Becker et al. [2003], Otto et al. [2006], Liu and Witelski [2020], Dallaston et al. [2021], the references therein, and Witelski [2020] for a recent overview.Potentials of the form (1.20) have also been successfully employed in the analysis of the macroscopic dynamics of wetting: see e.g.Eggers [2005a], Pismen and Eggers [2008], Savva and Kalliadasis [2011], and the references therein.However, this approach obviously can not capture pancake-shaped equilibria in dry complete wetting.
Remark 1.2.One may question whether a microscopic contact angle equal to π/2 is consistent with the small ratio of vertical vs horizontal length-scale required by lubrication approximation.In this respect, we should mention that the rigorous derivation of the thinfilm equation in Giacomelli and Otto [2003] only requires a global smallness conditions on such ratio, in form of relations between mass, energy, and second moments (the result is proved for Darcy's flow in complete wetting, but it is plausible that similar conclusions may be drawn as well for Stokes flow and partial wetting).In fact, the validity of lubrication theory under global (hence weak) assumptions is most evident when looking at the statics: for instance, in partial wetting (S < 0) with P ≡ 0 and one space dimension, it has been shown that under the sole assumptions that h ≥ 0 has finite energy, mass, and second moment, cf.Giacomelli and Otto [2001, (3) in Proposition 1].Note, however, that the finite-energy requirement is violated when m ≥ 3.This is another indication of the necessity for a cut-off at a molecular-size length-scale when van der Waals forces dominate as h → 0.

Travelling waves
As we mentioned, the idea that a film can spread because of a gradient of the disjoining pressure Π = −P is not new [Starov, 1983, de Gennes, 1984, Hervet and de Gennes, 1984].In Section IV.C.3 of his review, de Gennes [1985] presents a heuristic analysis of advancing travelling waves for (1.1) in the case of van der Waals potentials.We will now revisit part of his discussion in a more general way, i.e. assuming that and that The cases B > 0, resp.B < 0, correspond to long-range repulsive, resp.attractive, potentials.Once again, for m < 3 (2.1) may also be thought of as a cut-off of van der Waals potentials at molecular scales, such as in (1.19).Since our focus is on the contactline paradox, we adopt the no-slip condition, i.e. m(h) = 1 3 h 3 (in this respect, see also Remark 2.1 in §2.3).
We look for traveling-wave solutions to (1.1) with a constant speed V : We require H to display a contact line; capitalizing on translation invariance, we assume without losing generality that the contact line is located at y = 0: In addition, we ask that H connects to a bulk profile as y → +∞: Plugging the Ansatz (2.3) into (1.1) and using (1.3) yields, in case of (1.2), 3µV Assuming null mass flux through the contact line gives, after an integration, (2.5) Note that |U | coincides, up to a factor three, with the capillary number.Finally, we ask H to have finite rate of bulk dissipation near the contact line, in the sense that (cf.(1.7)) For brevity, we will call front a solution to (2.5)-(2.4)satisfying (2.6).Of course, as in the static case, it will be important to highlight which fronts have finite energy near the contact line, in the sense that (2.7) We will argue that singular potentials generically solve the contact-line paradox, in the following sense: (Q) Quadratic fronts.Assume (1.3), (1.2), (2.1), and (2.2).For any U ∈ R \ {0} there exists a two parameter (a > 0, b ∈ R), hence generic, family of fronts.They have quadratic growth as y → +∞, and satisfy where O(1), o(H m−1 ) and c 1 ∈ R are determined by m, U , P , a, and b.These fronts have finite energy if and only if m < 3.
This shows that, even if a no-slip condition is adopted at the contact-line, i.e. m(h) = 1 3 h 3 , singular potentials allow the existence of generic fronts (both advancing and receding) for any value of the normalized speed U .Since fronts have finite rate of dissipation, this means that singular potentials stand as an alternative solution to the contact-line paradox.In terms of H, (2.9) translates into which coincides with (1.14).Therefore, as in the static case, fronts are compatible with a fully consistent continuum theory down to h = 0 if m < 3, whereas if m ≥ 3 the energy is unbounded, and one must cut-off the fronts at a molecular length-scale .In this respect, (Q) parallels the static summary (1.18).
Note that (2.12) does imply linear-log behavior as y → +∞: as y → +∞. (2.13) As quadratic ones, also linear-log fronts are compatible with a fully consistent continuum theory down to h = 0 only if m < 3, whereas a molecular-sized cut-off is necessary if m ≥ 3.
Remark 2.1.Since our focus is on the contact-line paradox, we have only discussed the mobility m(h) = 1 3 h 3 , corresponding to the no-slip condition.However, analogous arguments yield (Q) and (L) also for mobilities of the form m(h In slippage models with P ≡ 0, a single linear-log front (and a one-parameter family of quadratic fronts) can be identified by imposing a condition on the value of the microscopic contact angle H y (0): in other words, the microscopic contact angle H y (0) may be taken as one of the parameters spanning the fronts.This additional condition on the contact angle is also necessary for uniqueness of generic solutions to (1.1) [Giacomelli et al., 2008, 2014, Knüpfer, 2011, 2015, Knüpfer and Masmoudi, 2013, 2015, Gnann, 2015, Gnann and Petrache, 2018].In (Q) and (L) above, the microscopic contact angle can not be a selection criterion, since all fronts have H y (0) = +∞.This points to the necessity of a different criterion which, for instance, singles out a unique linear-log front.
One possible selection criterion is the so-called maximal film, an advancing travelling wave supported in R (i.e., without a contact line).For van der Waals potentials, the maximal film has been successfully employed as a first approximation of droplets' advancing fronts for large positive spreading coefficient, where a prominent precursor region is supposed to form ahead of the macroscopic contact line.For instance, it is used by Hervet and de Gennes [1984] and de Gennes [1985] to infer the Voinov-Cox-Hocking logarithmic correction to Tanner's law [Voinov, 1976, Tanner, 1979, Cox, 1986, Hocking, 1983, 1992]; see e.g. the discussion in Eggers and Stone [2004].We will argue that such maximal film exists for any m ≥ 2: (M) Maximal film.Assume (1.3), (1.2), (2.1), and (2.2).For any U > 0 and m ≥ 2, there exists a unique (up to translations) solution H M to (2.5) in R such that H M (y) → 0 + as y → −∞ and (2.11) holds.They satisfy (2.13) and (2.14) In §2.1-2.3 we provide the formal asymptotic arguments which motivate (Q), (L) and (M); in §2.4 we give numerical examples supporting them; finally, in §2.5 we compare them with analogous results for the slippage model.
2.1.Asymptotics near the contact line.Note that (2.5) is autonomous: as customary, it is convenient to get rid of translation invariance by exchanging dependent and independent variable, thus reducing the order of the ODE.Therefore, we let (2.15) For H y > 0 in a neighborhood of H = 0, (2.5) reads as (2.16) At leading order as H → 0, a simple asymptotic expansion using (2.1) shows that two cases occur: (2.17) Case (a), resp.(b), may be read off from (2.16) by neglecting the first term on the right-hand side, resp.the left-hand side.
We anticipate that the solutions in (a) are generic and have finite rate of dissipation, whereas those in (b) are non-generic and have unbounded rate of dissipation.However, the solutions in (b) will capture the maximal film identified in (M).We thus distinguish the two cases.
2.1.1.Case (a).We linearize (2.16) around P (H): define the function v(H) as It follows from (2.16) that where The linearization of (2.18) around v = 0 is given by The left-hand side is an Euler equation, whereas the bracketed coefficients on the righthand side are o(1) as H → 0 + in view of (2.1).Therefore the equation has a twoparameter family of solutions of the form its regularity improves) is a function determined by U , P , m, c 1 , and c 2 .In terms of H, this translates into (2.9).Returning to the y variable, the additional degree of freedom coming from invariance under translation y → y − y 0 is spent to match the condition H(0) = 0. Therefore (2.9) translates into Since 2 m+1 < 1 for m > 1, these solutions have finite rate of dissipation; in addition, they have finite energy if and only if m < 3: In summary: (TW 0 ) Assume (1.3), (1.2), and (2.1).Locally for y 1, for any U ∈ R \ {0} there exists a two-parameter family of generic solutions H to (2.5)-(2.4a)satisfying (2.9) and (2.10).Their rate of bulk dissipation is finite, in the sense that (2.6) holds; if m < 3 their energy is also finite, in the sense that (2.7) holds.
(2.21b) In terms of H(y) as y → 0 + , for U > 0 and m < 2 (2.21) translates into a one-parameter, hence non-generic, family of solutions to (2.5)-(2.4a)which had already been identified by Bertozzi and Pugh [1994]: they all behave as Note the constraint m < 2: if m ≥ 2, H diverges as y → 0 + , hence it is not admissible.
(TW ∞ -Q) For any U ∈ R \ {0} there exists a generic, three-parameter (including translation) family of quadratically growing solutions: with a > 0 and b, y 0 ∈ R.
(TW ∞ -L) For any U > 0 there exists a non-generic, two-parameter (including translation) family of linear-log solutions satisfying (2.12) with a ∈ R.
To motivate (TW ∞ ), in view of (2.2) we rewrite (2.5) as (2.23) The asymptotic expansion yielding (TW ∞ -Q) is straightforward.For (TW ∞ -L), let U > 0. The equation for Following Giacomelli et al. [2016], we exploit the homogeneity of the (B = 0)-part of the equation letting s = log H, which yields This equation has been analysed in Giacomelli et al. [2016, Section 4 and 5], with a slippage-type perturbation (namely, f (s, u) = (1 + e (3−n)s ) −1 ) whose specific form is however immaterial as long as f (s) = O(s −2 log s).Their analysis shows that (2.24) with f = 0 has a one-parameter family of solutions such that u(s)/s → 1 and u (s) → 1 as s → +∞, with an asymptotic expansion of the form Since f (s, u) ≈ e s(1−p) u 1/3 and p > 1, it is apparent that f produces only an exponentially small perturbation: thus solutions to (2.24) have the same behavior, which in terms of H = e s yields (2.12).

Global behavior.
For the global picture, one has to make sure that local solutions are global.This is not always the case, in the sense that (2.5) also has generic solutions with compact support, a feature which is common to the slippage model.However, we have strong numerical evidence that generic members of both (TW ∞ -Q) and (TW ∞ -L) touch down to H = 0 at some point y 0 ∈ R (see §2.4).Capitalizing on translation invariance, one of the three parameters in (TW ∞ -Q) may be used to match H(0) = 0, yielding a two-parameter family of fronts satisfying (TW 0 ): this yields (Q).Analogously, one of the two parameters in (TW ∞ -L) may be used to match H(0) = 0, yielding a oneparameter family of fronts satisfying (TW 0 ): this yields (L).Finally, up to a translation, there exists a one parameter family of separatrices H sep emanating from −∞ as well as a one-parameter family of linear-log fronts emanating from +∞.Since (2.5) is of third order but autonomous, this entails a unique (up to translation) maximal film H M which satisfies (2.5) and is such that both (TW −∞ ) and (TW ∞ -L) hold: this yields (M).
2.4.Numerical observations.In order to provide numerical evidence of (Q), (L), and (M), we take as prototype example (2.25) The reason for considering such simple model instead of, for instance, (1.19) or (1.20), is twofold.Firstly, this choice is sufficient for a first numerical check on (Q), (L) and (M), since they quantitatively depend only on the behavior of P as h → 0 + (its sign and decay at infinity matters only qualitatively, independently of the power-law exponent p).Secondly, we can capitalize on the homogeneity of P to normalize the dimensionless speed U to ±1: indeed, letting we may rewrite (2.5) with (2.25) as Of course, a quantitative investigation of the fronts' behavior in the intermediate regions and/or in terms of the parameters, will require both a more careful choice of P and a more extensive numerical study, both outside the scope of this contribution.Note that a change in sign of U is equivalent, in (2.27), to a change of sign of ŷ, and that the left-hand side of (3.16) is unaffacted by the latter change: hence, removing hats, in place of (2.27) we will equivalently consider with the understanding that • advancing fronts (U > 0) correspond to solutions to (2.28) with H(0) = 0, supp H = [0, +∞) and H(+∞) = +∞; • receding fronts (U < 0) correspond to solutions to (2.28) with H(0) = 0, supp H = (−∞, 0] and H(−∞) = +∞.
The picture confirms the existence of both a two-parameter family of quadratic fronts (both advancing and receding, see (Q)) and two one-parameter families of advancing linear-log fronts (see (L)), resp.separatrices (see (TW −∞ )).In this case, the two parameters which span the fronts are taken to be α = H| Hyy=0 and β = H y | Hyy=0 .It is interesting to compare the shapes of the linear-log fronts H L for varying values of α = H L | (H L )yy=0 .The shapes reported in Fig. 4 show that H L increase as α increases.It also shows that a prominent precursor region forms ahead of the "macroscopic contact line" for small values of α. Figure 5 shows the maximal film H M (see (M)), which is obtained observing that (H M ) yy (y) → 0 as y → ±∞: hence there exists y 0 ∈ R such that (H M ) yyy (y 0 ) = 0, which implies that m(H M ) y (y 0 ) = H m−1 M (y 0 ).Then H M is identified by shooting from y 0 , using H M (y 0 ) and (H M ) yy (y 0 ) as parameters.2.5.Comparison with slippage models.It is useful to compare the features in (Q), (L) and (M) with parallel ones for the case P ≡ 0 under slip conditions.In this case, traveling wave solutions (if they exist) solve with the same boundary conditions.First of all, (Q) and (L) obviously contrasts (2.30) in the no-slip case λ = 0. Indeed, solutions to (2.30) with λ = 0 exist only if U < 0 (receding), but their rate of bulk dissipation density is not integrable near y = 0: indeed, as y → 0 + .Therefore, fronts of (2.30) do not exist at all if λ = 0. On the other hand, if λ > 0 (positive slippage), computations analogous to the ones above show that the picture is very much the same as in (Q) and (L): for any U ∈ R (U > 0 if H y (0) = 0 and n > 3 2 ) there exists a two-parameter family of quadratic fronts satisfying (2.22) (though with a different remainder); in addition, for any U > 0 there exists a one-parameter family of linear-log fronts satisfying (2.12)-(2.13)[Boatto et al., 1993, Buckingham et al., 2002, Chiricotto and Giacomelli, 2011, Giacomelli et al., 2016].A prototype case is given in Fig. 6: the only notable qualitative difference is that H sep are compactly supported; in fact, the unique separatrix with zero microscopic contact-angle coincides with the unique linear-log front in complete wetting, thus it is the counterpart of the maximal film H M in (M).

Thermodynamically consistent contact-line conditions
For exact, compactly supported solutions to the full evolution equation (1.1), the maximal film can obviously not be taken as a selection criterion for the fronts.In this section we will therefore identify a different criterion, which replaces contact-angle conditions in slippage models: it consists in a class of thermodynamically consistent contact-line conditions modelling friction at the contact line.The class will be identified by requiring dissipativity of the energy along the flow, in the spirit of the proposal by Ren and E [2007] (see also Ren et al. [2010], Ren and E [2011]).
We assume for simplicity that {h > 0} is and remains connected for all times (i.e.we exclude coalescence or splitting of droplets): s ± (t) denoting the contact lines.Since s ± (t) are unknown and (1.1) is of fourth order, three conditions are needed for well-posedness.Two of them are obvious: (3.1) The first one defines the contact lines s ± (t), while the second one is a kinematic condition guaranteeing no mass flux through s ± (t).The third condition, the so-called contact-line condition, is yet debated (to a certain extent inevitably, due to the variety of material properties and configurations, which may involve surface roughness and hysteretic effects; see e.g.Feldman and Kim [2018], Alberti andDeSimone [2005, 2011], as well as the above-mentioned reviews).The most common one amounts to prescribing a constant microscopic contact angle equal to the static one, as defined by (1.9): Of particular interest to us is a relatively recent proposal by Ren and E [2007] (see also Ren et al. [2010], Ren and E [2011]), based on consistency with the second law of thermodynamics, which also gives a robust motivation to older models by Greenspan [1978] and Ehrhard and Davis [1991].In lubrication approximation [Chiricotto andGiacomelli, 2011, 2013], when P ≡ 0 and S = − 1 2 tan 2 θ S (the moist case, cf.(1.9)), the simplest form of the Ren-E model reads as follows3 : with µ cl > 0 a coefficient measuring friction at the contact line.Indeed, under (3.3), the energy balance reads as (see Chiricotto and Giacomelli [2017]), which shows that (3.3) is consistent with the second law and accounts for frictional forces at the contact line, with µ cl ≥ 0 a friction coefficient.When µ cl = 0 (null contact-line friction), (3.3) coincides with (3.2).
We will now revisit the argument for (3.3) in the case of a singular potential P .We base our computations on the expectation that the behavior of generic solutions coincides with that of the fronts near the contact line: letting we assume that (2.9) = O(1) as y → 0 + , (3.5d) We now notice three facts.Firstly, and crucially, the first boundary term remains bounded as ε → 0: indeed, = O(1) as ε → 0. (3.8) In addition, the second boundary term vanishes as ε → 0: Finally, the integral on the right-hand side of (3.7) is finite: indeed, which is integrable at x = s ± (t) since 2 m+1 < 1. Passing to the limit as ε → 0, we obtain from (3.6)-(3.9)that (3.10) Remark 3.1.This formal computation is fully consistent if m < 3.If m ≥ 3, instead, the limit ε → 0 on the left-hand side of (3.10) does not make sense since E[h] ≡ +∞ in that case.However, (3.7) does make sense for any positive ε, and the limit as ε → 0 on its right-hand side makes sense for any m > 1.
In order to be thermodinamically consistent, the contact-line condition has to be such that the free energy is dissipated along the flow, i.e., that the r.h.s. of (3.10) is nonpositive: this leads to the following class of contact-line conditions: The term f ( ṡ) is a contribution to the dissipation which is concentrated at the contact line: in the description of Bonn et al. [2009], it corresponds to the term W m (U ) in formula (75).Recalling that Q(h) = P (h) − S, we see that the spreading coefficient S enters the contact-line conditions (3.11)-(3.12) in an essential way, in analogy with the contact-line conditions (3.2)-(3.3)for the slippage model.
The simplest choice of f is a linear relation f ( ṡ) = µ cl ṡ (µ cl ≥ 0), in analogy with (3.3); it leads to (3.12) Substituting (3.12) into (3.10)we obtain an energy balance analogous to the one in (3.4): which encodes a quadratic dissipation of kinetic energy through frictional forces acting at the contact line.Note that (3.13) coincides with (1.7) if µ cl = 0.
Let us comment on the contact-line condition (3.12).Its left-hand side is zero for the global minimizers h min discussed in §1.3 (see Durastanti and Giacomelli [2022, Theorem 4.5]).Also, its left-hand side is well defined on the fronts.Indeed, though both summands are unbounded as x → s ± (t), their difference is not: 3),(2.1),(2.9) For traveling waves with constant speed V = γ 3µ U , s − (t) = −V t and the contact-line condition (3.12) reads as (1) For any U ∈ R \ {0}, (1.1) has a one-parameter family of quadratic fronts H (see (Q)) such that (3.15) holds; (2) For any U > 0, (1.1) has a unique linear-log front H L (see (L)) such that (3.15) holds.
To support the choices of both (3.12) as free boundary condition and Θ[H] as selection parameter, we report numerical values of Θ[H L ] as computed for the rescaled equation (2.28): under (2.26), the rescaled version of (3.15) is, after removing hats, It is apparent from Fig. 7(A) that Θ[H L ] monotonically covers the whole real line as linear-log solutions are spanned: in particular, a unique linear-log front can be selected such that the contact-line condition (3.16) holds.This confirms the expectation in (S2).
For completeness, in Fig. 7(B) we also report numerical values of the separatrix H sep which discriminates between receding and compactly supported solutions.It is apparent that Θ[H sep ] increases with α and diverges to −∞ as α → 0 + .Combining Fig. 7(A) with Fig. 4, it is also apparent that, as Θ[H L ] decreases, a more prominent precursor region forms ahead of the macroscopic contact line (Fig. 8).Thus, if (3.16) is assumed as a contact-line condition, we expect that H L matches the following intuitive properties: • for fixed µ cl and S, a greater speed U yields steeper profiles of H L ; • for fixed U and S, a greater contact-line friction µ cl yields steeper profiles of H L ; • for fixed µ cl and U , a larger positive spreading coefficient S yields gentler profiles of H L .In particular, under the contact-line condition (3.16), numerics suggest that larger positive spreading coefficients S yield more prominent precursor regions ahead of the macroscopic contact line (Fig. 8).This agrees with the discussion in de Gennes [1985] for m = 3, which is instead based on perturbations of the maximal film.diverge to −∞ as H| Hyy=0 → 0, (Fig. 7), Θ[H] covers the whole real line as advancing, resp.receding, traveling waves are spanned, thus confirming (S1).There is, however, a difference between advancing and receding fronts, since receding ones may be nonmonotonic (Figg. 2 and 3); note that the same happens for slippage models (Fig. 6).This reflects into a lack of monotonicity of Θ[H] for fixed α on receding waves.Note that the minimum of Θ[H] appears to be on the left half-plane, hence the corresponding waves have negative derivative at their inflection point, thus they are monotonic.Therefore the branch of receding fronts emanating from Θ[H] = +∞ consists of monotone ones.This phenomenon might be related to the existence of a "limiting speed" of receding fronts calculated by Eggers [2004Eggers [ , 2005b] ] in the slippage case.

Conclusions and open questions
We have discussed thin-film models under singular potentials: 2) Based on formal arguments supported by numerical evidence, we have argued that singular potentials generically solve the contact-line paradox, in the sense that (4.1) admits for any value of m > 1: (Q) a two-parameter family of both advancing and receding traveling-waves with finite rate of dissipation at the contact line; (L) a one-parameter family of advancing "linear-log" travelling waves displaying a logarithmically corrected linear behavior in the bulk, also with finite rate of dissipation at the contact line.
In agreement with mass-constrained steady states, travelling waves have finite energy if and only if m < 3, whereas for m ≥ 3 a cut-off at a molecular-size length-scale is necessary (cf.e.g.(1.19)).However, the qualitative properties of the waves are the same for any m > 1.Our formal arguments also suggest that for m ≥ 2 and any positive speed there exists: (M) a unique maximal film, i.e., an advancing linear-log travelling wave H(y) which decays to zero as y → −∞ instead of touching down to zero at a finite point.
Intermolecular potentials thus stand as a possible solution to the contact-line paradox, alternative to the most common one, given by slippage models: For equation (4.3), the classification of travelling waves is analogous to (Q) and (L) (cf.§2.5); however, the microscopic contact angle h x | {h=0} may be used as a parameter to span them, thus selecting a unique advancing linear-log front.This is impossible for (4.1)-(4.2),since in that case the microscopic contact angle is always π/2.Here, we have also proposed a class of thermodynamically consistent contact-line conditions, which replaces contact-angle ones and is expected to single out a unique advancing linear-log front with a contact line.The simplest among such conditions reads as Θ[h(t)] := 1 2 h 2 x − P (h) | x=s ± (t) = ± µcl γ ṡ± (t) − S, (4.4) where {h(t) > 0} = (s − (t), s + (t)), µ cl ≥ 0 is a contact-line frictional coefficient, and S is the non-dimensional spreading coefficient.We expect that: (S) for any value of the speed, (4.4) selects a one-parameter family of fronts in (Q) and a unique linear-log front in (L).
Numerical evidence also suggests that linear-log fronts are steep for Θ[h] 1, whereas they display a precursor region ahead of the macroscopic contact line for Θ[h] −1.Therefore, we expect that (4.4) yields precursor regions for large positive values of the spreading coefficient.
The above four observations issue quite a few challenges.
• The rigorous validation of (Q), (L) and (M) is highly desirable.Once the asymptotics in (Q) and (L) have been proved, we expect that (S) will follow as a byproduct.• For relatively large values of S, it would be interesting to quantify height and length of the precursor region, which appears to exist ahead of the "macroscopic" contact line, in relation to the contact-line condition (4.4).• It would be very useful to have numerical simulations and/or matched asymptotic studies available for generic solutions to (4.1) with the contact-line condition (4.4), for potentials P of the general form (4.2).Of particular interest would be the (in)stability of advancing/receding traveling waves, the detection of scaling laws -such as the Voinov-Cox-Hocking logarithmic correction to Tanner's law-, an estimate of the rate of convergence to equilibria, and an insight on the evolution of the precursor region for relatively large values of S. • Based on (S), for potentials P of the form (4.2), we conjecture that for any nonnegative h 0 ∈ H 1 (R) such that E[h 0 ] < +∞ there exists a unique solution to (4.1) with the contact-line conditions (3.1) and (4.4).A difficult but extremely interesting task would be to develop a well-posedness theory at least when h 0 is a perturbation of a traveling wave, in the spirit of [Giacomelli et al., 2008, 2014, Knüpfer, 2011, 2015, Knüpfer and Masmoudi, 2013, 2015, Gnann, 2015, Gnann and Petrache, 2018].• De Silva and Savin [2022] recently analyzed the Γ-convergence of the energy E in the limit m → 3 − and vanishing A (A = A m ≈ (3 − m) 2 → 0 as m → 3).It would be interesting to understand how this scaling limit extends to the dynamic framework.• In relation to the issue of infinite energy for m ≥ 3, it would be interesting to explore the effect of taking the full curvature operator into account, i.e., replacing 1 + 1 2 h 2 x with 1 + h 2 x .We are aware of only a few studies [Novick-Cohen, 1992, 1993, Minkov and Novick-Cohen, 2001, 2006], dealing with the statics under convex potentials.
• Interesting, though of seemingly lesser impact, would also be to face the above challenges for more general mobilities, e.g. of the form m(h) = 1 3 (h 3 + λ 3−n h n ) (cf. alos Remark 2.1).

Appendix
The limitation (1.16) of the energy in (1.6) is well known, though with slightly different formulations; see e.g.Durastanti and Giacomelli [2022, Lemma 2.10] in R and Lazer and McKenna [1991, Theorem 2] on bounded domains.For completeness, here we provide a comprehensive statement in R. Note that E is a singular version of the Alt and Phillips [1986] functional.

Figure 2 .
Figure 2. Solutions to (2.29) with m = 2, α = 1/4 (top) and α = 1 (bottom), at two different scales.For α = 1/4 (top), the separatrix H sep and the black receding front are indistinguishable for small heights, as well as the gray compactly supported solution and the linear-log front.