Molecular-level theory of oscillatory solvation forces and the classification of wetting regimes on smooth solids

The continuum analysis of Brochard-Wyart, Di Meglio, Quéré and de Gennes (Langmuir 7, 335, 1991) classified the static wetting of a nonvolatile liquid on a smooth solid by the signs of the spreading coefficient $S$ and the effective Hamaker constant $A$, assuming a smooth effective interface potential $F(e)$ (the free energy per unit area of a flat liquid film of thickness $e$) that interpolates between the short-range boundary $P(0)=S$ (where $P(e)$ is the thickness-dependent interaction part of $F(e)$) and the long-range van der Waals tail $A/12\pi e^2$. This yields complete wetting (a pancake of thickness $e_s=(A/4\pi S)^{1/2}$), pseudo-partial wetting (a finite-angle droplet coexisting with a film of thickness $e_m$ where $F(e)$ is minimal), and partial wetting on a dry solid. The authors explicitly ignored oscillations of $F(e)$ at molecular thicknesses, produced by molecular packing (solvation forces, observed as molecular layering in spreading films), which may alter the wetting criteria, particularly for $A>0$, where oscillatory minima could convert predicted complete wetting into pseudo-partial wetting. Later work established pseudo-partial wetting experimentally, confirmed that molecular ordering and oscillatory structural forces influence wetting, and resolved nanoscale contact-line structure for model fluids by density-functional and molecular-dynamics methods; but no molecular-level derivation of the oscillatory $F(e)$ for realistic solid-liquid pairs and its consequences for the global wetting classification is known. The open problem is to build that theory: derive $F(e)$ with its molecular-scale oscillatory structure, determine how it shifts the regime boundaries (including the $A>0$ pseudo-partial case), resolve the sub-continuum structure of molecular films and contact-line tips, and thereby complete the classification of wetting regimes.

Background

Wetting is the statics of how a liquid meets a solid: a sessile droplet either forms a spherical cap with a finite contact angle $\theta$ (partial wetting) or spreads into a film covering the whole surface (complete wetting). In partial wetting, $\theta$ obeys Young's equation $\gamma_{so}=\gamma_{sl}+\gamma\cos\theta$, where $\gamma_{so}$, $\gamma_{sl}$ and $\gamma$ are the solid-vapor, solid-liquid and liquid-vapor interfacial tensions. The spreading coefficient $S=\gamma_{so}-\gamma_{sl}-\gamma$ is the free-energy change per unit area when dry solid is replaced by bulk liquid: $S>0$ favors complete wetting, $S<0$ partial wetting.

For films of intermediate thickness $e$ the thermodynamics is governed by the effective interface potential $F(e)=\gamma_{sl}+\gamma+P(e)$, the free energy per unit area of a flat film, where $P(e)$ collects the intermolecular interactions between the film's two boundaries; its negative derivative $\Pi(e)=-dP/de$ is the disjoining pressure, the extra pressure a thin film exerts because of surface forces. At large $e$, dispersion (van der Waals) forces dominate, $P(e)=A/12\pi e^2$, where $A$ is an effective Hamaker constant (the standard amplitude of the van der Waals interaction across a medium) built from the difference of solid-liquid and liquid-liquid dispersion couplings; as $e\to 0$, $P(0)=S$ absorbs all short-range chemical contributions (dipoles, hydrogen bonding, density modification near the wall). A key point of the modern theory initiated by de Gennes (1985) is that $S$ and $A$ are essentially independent parameters, so a consistent classification must use both.

Brochard-Wyart, Di Meglio, Quéré and de Gennes (1991) completed this continuum classification for nonvolatile liquids, assuming $F(e)$ is smooth and non-oscillatory. (i) For $S>0$, $A>0$ the droplet spreads completely into a pancake of thickness $e_s=(A/4\pi S)^{1/2}=a(3\gamma/2S)^{1/2}$, terminating with a parabolic tip $z^2=2a(x-x_L)$ on the dry solid. (ii) If $F(e)$ has an absolute minimum at a finite thickness $e_m$ (realized when $A<0$ combines with suitable short-range contributions, for either sign of $S$), a large droplet does not spread completely: a spherical cap with $\cos\theta=1-a^2/2e_m^2$ coexists with a film of thickness $e_m$ covering the solid. This is the pseudo-partial wetting regime, which accounts for silicone-oil droplets on silicon that leave a molecular precursor film while keeping a finite contact angle (Silberzan and Léger, 1991). (iii) Otherwise partial wetting holds on a strictly dry solid, with Young's angle and hyperbolic profiles $z^2=(\theta x)^2\mp a^2/\theta^2$ near the contact line. Here $z(x)$ is the interface height, $x_L$ the contact-line position, and $a$ a microscopic length defined by $|A|/6\pi=\gamma a^2$.

Two molecular-scale effects lie outside this picture. First, ellipsometry (an optical technique that measures film thickness with sub-molecular resolution) showed, for spreading droplets on silicon (Heslot, Fraysse and Cazabat, 1989), that even simple liquids advance as a sequence of discrete molecular layers, and surface-force measurements have long established that liquids confined near very smooth walls develop oscillatory solvation (structural) forces, an image of hard-sphere correlation functions reflecting molecular packing. An oscillatory $F(e)$ can carry additional minima at thicknesses of a few molecular sizes $a_0$. The 1991 analysis explicitly ignores such oscillations of large amplitude, noting that they may alter the wetting criteria, particularly for $A>0$, where oscillatory minima could create a pseudo-partial-wetting-like state although the long-range forces alone predict complete wetting; for $A<0$ the conclusions are less affected. Second, the continuum description itself breaks down at thicknesses of order $a_0$, so it cannot describe the detailed structure of the molecular tip of the wedge or pancake in complete wetting, nor of the film in pseudo-partial wetting.

The open problem is to bring this classification down to the molecular scale: to derive $F(e)$, with its oscillatory short-range structure, from molecular interactions, to work out how the oscillations shift the regime boundaries, and to resolve the sub-continuum structure of the films and tips that the continuum theory can only treat as boundary conditions.

Problem Statement

Develop a molecular-level theory of the static wetting of nonvolatile liquids on smooth solids that resolves the role of the short-range oscillatory (solvation/structural) forces left open by the continuum analysis of Brochard-Wyart, Di Meglio, Quéré and de Gennes (1991). Concretely: (i) derive, from molecular interactions (by analytical theory, classical density-functional theory - the statistical-mechanical framework that obtains equilibrium fluid density profiles and free energies by minimizing a functional of the local density - or validated molecular simulation), the effective interface potential $F(e)$ - equivalently the disjoining pressure $\Pi(e)$ - including its molecular-scale oscillatory structure for representative solid-liquid pairs, in place of the smooth interpolation between the short-range boundary $P(0)=S$ (with $P(e)$ the interaction part of $F(e)$) and the van der Waals tail $A/12\pi e^2$ assumed in the continuum treatment; (ii) determine how oscillatory minima of $F(e)$ modify the wetting criteria - in particular whether and under what conditions they stabilize pseudo-partial wetting (a finite-angle droplet coexisting with a molecular film) in systems where the long-range forces alone ($A>0$) predict complete wetting - and thereby establish the corrected boundaries of the wetting-regime classification in terms of the underlying molecular parameters rather than only the phenomenological signs of $S$ and $A$; and (iii) resolve the sub-continuum structure of the molecular film in pseudo-partial wetting and of the molecular tip of the spreading wedge or pancake in complete wetting. The theory is to give predictions for specific material systems that can be tested experimentally or by molecular simulation.

References

1. F. Brochard-Wyart, J. M. Di Meglio, D. Quéré, P. G. de Gennes, 'Spreading of nonvolatile liquids in a continuum picture', Langmuir 7(2), 335-338 (1991). DOI: 10.1021/la00050a023, https://doi.org/10.1021/la00050a023
2. P. G. de Gennes, 'Wetting: statics and dynamics', Reviews of Modern Physics 57, 827-863 (1985). DOI: 10.1103/RevModPhys.57.827, https://doi.org/10.1103/RevModPhys.57.827
3. F. Heslot, N. Fraysse, A. M. Cazabat, 'Molecular layering in the spreading of wetting liquid drops', Nature 338, 640-642 (1989). DOI: 10.1038/338640a0, https://doi.org/10.1038/338640a0
4. P. Silberzan, L. Léger, 'Evidence for a new spreading regime between partial and total wetting', Physical Review Letters 66, 185-188 (1991). DOI: 10.1103/PhysRevLett.66.185, https://doi.org/10.1103/PhysRevLett.66.185
5. R. Pandit, M. Schick, M. Wortis, 'Systematics of multilayer adsorption phenomena on attractive substrates', Physical Review B 26, 5112-5140 (1982). DOI: 10.1103/PhysRevB.26.5112, https://doi.org/10.1103/PhysRevB.26.5112
6. C. Merkl, T. Pfohl, H. Riegler, 'Influence of the molecular ordering on the wetting of SiO2/air interfaces by alkanes', Physical Review Letters 79, 4625-4628 (1997). DOI: 10.1103/PhysRevLett.79.4625, https://doi.org/10.1103/PhysRevLett.79.4625
7. L. Esibov, D. Sarkisov, U.-S. Jeng, M. L. Crow, A. Steyerl, 'Dynamics of pseudo-partial wetting studied by neutron reflectometry', Physica B 241-243, 1077-1079 (1997). DOI: 10.1016/S0921-4526(97)00796-5, https://doi.org/10.1016/S0921-4526(97)00796-5
8. T. Getta, S. Dietrich, 'Line tension between fluid phases and a substrate', Physical Review E 57, 655-671 (1998). DOI: 10.1103/PhysRevE.57.655, https://doi.org/10.1103/PhysRevE.57.655
9. D. Bonn, E. Bertrand, N. Shahidzadeh, K. Ragil, H. T. Dobbs, A. I. Posazhennikova, D. Broseta, J. Meunier, J. O. Indekeu, 'Complex wetting phenomena in liquid mixtures: frustrated-complete wetting and competing intermolecular forces', Journal of Physics: Condensed Matter 13, 4903-4914 (2001). DOI: 10.1088/0953-8984/13/21/317, https://doi.org/10.1088/0953-8984/13/21/317
10. L. Boinovich, A. Emelyanenko, 'Wetting and surface forces', Advances in Colloid and Interface Science 165, 60-69 (2011). DOI: 10.1016/j.cis.2011.03.002, https://doi.org/10.1016/j.cis.2011.03.002
11. K. Koga, J. O. Indekeu, B. Widom, 'Infinite-order transitions in density-functional models of wetting', Physical Review Letters 104, 036101 (2010). DOI: 10.1103/PhysRevLett.104.036101, https://doi.org/10.1103/PhysRevLett.104.036101
12. P. Hopkins, M. Schmidt, 'First-order layering and critical wetting transitions in nonadditive hard-sphere mixtures', Physical Review E 83, 050602 (2011). DOI: 10.1103/PhysRevE.83.050602, https://doi.org/10.1103/PhysRevE.83.050602
13. N. Tretyakov, M. Müller, D. Todorova, U. Thiele, 'Parameter passing between molecular dynamics and continuum models for droplets on solid substrates: the static case', The Journal of Chemical Physics 138, 064905 (2013). DOI: 10.1063/1.4790581, https://doi.org/10.1063/1.4790581
14. L. Du, H. Bodiguel, C. Cottin, A. Colin, 'Dynamics of liquid-liquid menisci in pseudo-partial wetting', Chemical Engineering and Processing 68, 3-6 (2013). DOI: 10.1016/j.cep.2012.06.014, https://doi.org/10.1016/j.cep.2012.06.014
15. A. Nold, D. N. Sibley, B. D. Goddard, S. Kalliadasis, 'Fluid structure in the immediate vicinity of an equilibrium three-phase contact line and assessment of disjoining pressure models using density functional theory', Physics of Fluids 26, 072001 (2014). DOI: 10.1063/1.4886128, https://doi.org/10.1063/1.4886128
16. B. B. Straumal, I. Konyashin, B. Ries, A. B. Straumal, A. A. Mazilkin, K. I. Kolesnikova, A. M. Gusak, B. Baretzky, 'Pseudopartial wetting of WC/WC grain boundaries in cemented carbides', Materials Letters 147, 105-108 (2015). DOI: 10.1016/j.matlet.2015.02.029, https://doi.org/10.1016/j.matlet.2015.02.029
17. A. Nold, L. G. MacDowell, D. N. Sibley, B. D. Goddard, S. Kalliadasis, 'The vicinity of an equilibrium three-phase contact line using density-functional theory: density profiles normal to the fluid interface', Molecular Physics 116, 2239-2243 (2018). DOI: 10.1080/00268976.2018.1471223, https://doi.org/10.1080/00268976.2018.1471223
18. D. Bonn, J. Eggers, J. Indekeu, J. Meunier, E. Rolley, 'Wetting and spreading', Reviews of Modern Physics 81, 739-805 (2009). DOI: 10.1103/RevModPhys.81.739, https://doi.org/10.1103/RevModPhys.81.739
19. A. Nepomnyashchy, 'Droplet on a liquid substrate: wetting, dewetting, dynamics, instabilities', Current Opinion in Colloid and Interface Science 51, 101398 (2021). DOI: 10.1016/j.cocis.2020.101398, https://doi.org/10.1016/j.cocis.2020.101398
20. J. Lee, A. D. Nikolov, D. Wasan, 'Oscillatory structural forces probed by the nanofilm Macroscopic contact angle', Journal of Molecular Liquids 364, 119958 (2022). DOI: 10.1016/j.molliq.2022.119958, https://doi.org/10.1016/j.molliq.2022.119958
21. A. D. Nikolov, D. T. Wasan, P. Wu, 'Solvation forces versus the nano-colloidal structural forces under the film confinement: layer to in-layer structural transition in wetting solids', Current Opinion in Colloid and Interface Science 57, 101539 (2022). DOI: 10.1016/j.cocis.2021.101539, https://doi.org/10.1016/j.cocis.2021.101539
22. F. Oktasendra, A. Jusufi, A. R. Konicek, M. S. Yeganeh, J. R. Panter, H. Kusumaatmaja, 'Phase field simulation of liquid filling on grooved surfaces for complete, partial, and pseudo-partial wetting cases', The Journal of Chemical Physics 158, 204501 (2023). DOI: 10.1063/5.0144886, https://doi.org/10.1063/5.0144886 (preprint: arXiv:2305.02779, https://arxiv.org/abs/2305.02779)
23. A. Nold, B. D. Goddard, D. N. Sibley, S. Kalliadasis, 'Hydrodynamic density-functional theory for the moving contact-line problem reveals fluid structure and emergence of a spatially distinct pattern', Physical Review Fluids 9, 124003 (2024). DOI: 10.1103/PhysRevFluids.9.124003, https://doi.org/10.1103/PhysRevFluids.9.124003