Enhanced dip coating on a soft substrate

A solid, withdrawn from a wetting liquid bath, entrains a thin liquid film. This simple process, first described by Landau, Levich, and Derjaguin (LLD), is commonly observed in everyday life. It also plays a central role in liquid capture by animals, and is widely used for surface-coating purposes in industry. Motivated by the emerging interest in the mechanics of very soft materials, and in particular the resulting elastocapillary coupling, we develop a dip-coating model that accounts for the additional presence of a soft solid layer atop the rigid plate. The elastic response of this soft layer is described by a Winkler's foundation. Using a combination of numerical, scaling, and asymptotic-matching methods, we find a new softness-dependent power-law regime for the thickness of entrained liquid at a small capillary number, which corresponds to a modified physics at play in the dynamic meniscus. The crossover between this regime and the classical dip-coating one occurs when the substrate's deformation is comparable to the thickness of the entrained liquid film.

Figure 1

Schematic of the soft dip-coating problem. The inset exhibits a zoom in the dynamic-meniscus zone where the flow is localized. The liquid-air interfacial profile is denoted $h\left(z\right)$ and its curvature induces a negative Laplace pressure that generates a deformation $\delta \left(z\right)$ of the soft substrate.

Figure 2

Normalized thickness of the entrained liquid film versus capillary number, for different values of the dimensionless softness parameter $\mathcal{L}=\rho gt/E^{*}$, as obtained from the numerical integration of Eq. (5). The black dotted line represents the classical LLD law of Eq. (6) and the gray dashed line shows the soft-LLD law of Eq. (7). The slope triangles indicate the power-law exponents of Eqs. (6) and (7). The inset shows the thickness of the entrained film normalized by the LLD scaling versus the dimensionless parameter ${\mathrm{Ca}}^{2/3}/\mathcal{L}$.

Figure 3

Profiles of the liquid-air interface (blue) and the normal deformation of the elastic layer (green) normalized by the capillary length, as functions of the vertical position normalized by the capillary length, as obtained from the numerical integration of Eq. (5). The dimensionless softness parameter is set to $\mathcal{L}=0.1$. The capillary numbers are $\mathrm{Ca}={10}^{-1},{10}^{-2},{10}^{-3}$, and ${10}^{-4}$, in (a)–(d), respectively.

Figure 4

(a) Schematic showing the four regions of interest: (i) at large $z$, the entrained liquid film is flat and the elastic layer is not deformed; (ii) reducing $z$, one enters the elastocapillary region, with a constant fluid thickness and a progressively deformed elastic layer; (iii) a dynamic meniscus (“boundary layer”) then connects the elastocapillary region to the static meniscus; (iv) the static meniscus. Note that the schematic is reoriented with respect to Fig. 1. (b) Profiles of the liquid-air interface (blue) and the normal deformation of the elastic layer (green) normalized by the typical normal deformation of the soft layer [see Eq. (8)], as functions of the vertical position rescaled by the capillary length, as obtained from the numerical integration of Eq. (5) with $\mathrm{Ca}={10}^{-9}$ and $\mathcal{L}=0.01$. The dotted pink line corresponds to the outer elastocapillary solution given by Eq. (10), where the boundary layer position is indicated with a vertical dashed line. The inset shows a zoom of the boundary layer region, where the black dashed line displays the boundary-layer solution of Eq. (12). In the inset, the profiles are plotted with the self-similar variables.