Contact angles for perfectly wetting pure liquids evaporating into air: Between de Gennes-type and other classical models

The present theoretical study is concerned with evaporation-induced apparent contact angles for a perfectly wetting one-component liquid placed on a flat solid substrate and undergoing diffusion-limited evaporation into ambient air. The analysis pertains to a distinguished small vicinity of the contact line (the “microregion”), where such angles are established and where various microscopic effects typically enable relaxing the well-known evaporation-flux singularity. We proceed from a Joanny-Hervet-de Gennes-type approach, involving the spreading coefficient, disjoining pressure in the form of an inverse cubic law, and a truncated microfilm (precursor film) starting abruptly at a solid surface. A more classical regime with an (infinitely) extended adsorbed microfilm is recovered therefrom in the limit of large spreading coefficients upon additional incorporation of the Kelvin effect (dew point shift due to the liquid–gas pressure difference). The latter regime is critically revisited with a view to clarifying the scaling prefactor known in the literature. The influence of the kinetic resistance to evaporation is analyzed as well.

Figure 1

Sketch of the microregion profiles with truncated and extended microfilms.

Figure 2

Solid line: rescaled evaporation-induced contact angle ${\vartheta }_{\mathrm{mic}}$, which is actually the prefactor in a classical scaling law given by Eqs. (1, 2, 3) for the evaporation-induced contact angle, computed as a function of the dimensionless spreading coefficient $\mathrm{\Sigma }$, Eq. (4), in the framework of the basic model. Dashed line: Young's angle in the same scaling for comparison (defined only for $\mathrm{\Sigma }<0$—partial wetting). Inset: details near the minimum of the curve.

Figure 3

Computed dimensionless microregion film profiles $h\left(x\right)$ at different values of the dimensionless spreading coefficient ($\mathrm{\Sigma }=0,5\mathrm{and}20$) in the framework of the basic model. For comparative purposes, each profile has been shifted so as to have $h=5$ at $x=0$. The scales are defined in Eq. (11).

Figure 4

Geometric sketch of the vapor diffusion problem in the gas. The liquid layer is of negligible thickness at the scale of the gas phase and corresponds to $\phi =0$ in terms of the polar coordinates $\{r,\phi \}$ centered at the tip of the liquid layer ($\phi =\pi$ at the bare solid surface).

Figure 5

Solid lines: computed dimensionless evaporation flux distribution $j\left(x\right)$ along the microregion for a finite kinetic resistance to evaporation ($K=1$ and $K=5$) in the framework of the basic model $+$ kinetic resistance. Dashed line: the same distribution in the absence of kinetic resistance ($K=0$), implying a local thermodynamic equilibrium throughout, as given by Eq. (13) and used in the basic model. The scales are defined in Eq. (11) while the parameter $K$ in Eq. (27). At any $K$, we have $j\left(x\right)=K^{-1/2}{j}_1(x/K)$, where $j_1\equiv j$ at $K=1$ shown in the plot.

Figure 6

The same as in Fig. 2, but now in the framework of the basic model $+$ kinetic resistance for a few values of the kinetic resistance number $K=0,1,\mathrm{and}5$. Inset: ${\vartheta }_{\mathrm{mic}}$ versus $K$ for $\mathrm{\Sigma }=0$. The result for $K=0$ exactly coincides with the one of Fig. 2.

Figure 7

The same as in Fig. 2, but now in the framework of the basic model $+$ Kelvin effect for a number of values of the Kelvin number $\mathrm{Ke}=0,0.01,0.03,0.1,0.3\mathrm{and}1$. $\mathrm{Ke}$ is defined in Eq. (31) and related as $\mathcal{L}=3\times 2^{4/9}{\mathrm{Ke}}^{4/3}$ to the Laplace parameter $\mathcal{L}$ used in Refs. [28, 29]. The result for $\mathrm{Ke}=0$ exactly coincides with the one of Fig. 2.

Figure 8

Solid lines: computed dimensionless microregion film profiles $h\left(x\right)$ (left) and evaporation flux profiles $j\left(x\right)$ (right) within the basic model $+$ Kelvin effect at $\mathrm{Ke}=0.3$ for a number of values $\mathrm{\Sigma }=4,5,6\mathrm{and}7$. Dashed lines: extended-microfilm asymptotics. All profiles except for $\mathrm{\Sigma }=7$ are suitably $x$-shifted so as to highlight their actual partial overlapping. Horizontal thin solid lines (left) indicate the pancake thickness ${\mathrm{\Sigma }}^{-1/2}$ for some film profiles. The scales are defined in (11). For comparison, the film profiles shown in Fig. 3 correspond to $\mathrm{Ke}=0$ in current terms.

Figure 9

Rescaled evaporation-induced contact angle ${\vartheta }_{\mathrm{mic}}$, the prefactor in a classical scaling law given by Eqs. (1, 2, 3) for the evaporation-induced contact angle, versus the Kelvin number $\mathrm{Ke}$ in the microregion regime with an extended adsorbed microfilm (no kinetic resistance yet considered, $K=0$). Solid line: our present computation. Short-dashed line: asymptotic result in the limit $\mathrm{Ke}\ll 1$ obtained in Ref. [28] and also used in Ref. [29]. Long-dashed line: ditto but now also including a higher-order correction obtained in Appendix pp2. Dot-dashed line: asymptotic result in the limit $\mathrm{Ke}\gg 1$ adapted from Ref. [40].

Figure 10

Computed dimensionless microregion profiles (solid lines) of the film slope $h^{\prime }\left(x\right)$ for $\mathrm{Ke}=0.0001,0.001,0.01,0.1,\text{and}1$ (left) and evaporation flux $j\left(x\right)$ for $\mathrm{Ke}=0,0.03,0.1,0.3,\text{and}1$ (right). Regime with an extended adsorbed microfilm (and no kinetic resistance, $K=0$). The scales are defined in Eq. (11). The maximum values of both $h^{\prime }$ and $j$ decrease with increasing $\mathrm{Ke}$, which permits to establish an unambiguous correspondence between a given profile and its $\mathrm{Ke}$ value. Asymptotic behaviors given by Eqs. (39) and (41) (dashed lines).

Figure 11

Dependence on the kinetic resistance in the extended-microfilm regime. The result for $K=0$ (no kinetic resistance) reproduces the solid curve of Fig. 9, and its counterparts for $K=1\mathrm{and}5$ are also represented. Inset: ${\vartheta }_{\mathrm{mic}}$ versus $K$ at $\mathrm{Ke}=0.01$.

Figure 12

Solid lines: computed dimensionless microregion film profiles $h\left(x\right)$ exactly corresponding to the slope profiles of Fig. 10 (left). Larger $\mathrm{Ke}$ correspond to larger $h$ within the region shown. Dashed lines: for each case, the asymptotic behaviors given by Eq. (34) valid at large negative $x$ (indiscernible and not shown for smaller $\mathrm{Ke}$) and Eq. (A1) valid at large positive $x$ with the free coefficients computed from the full solution. Here Eq. (A1) has been extended up to and including the order $O\left(x^{-1}\right)$.