Asymptotic analysis of the contact-line microregion for a perfectly wetting volatile liquid in a pure-vapor atmosphere

We revisit the Wayner problem of the microregion of a contact line at rest formed by a perfectly wetting single-component liquid on an isothermal superheated flat substrate in an atmosphere of its own pure vapor. The focus is on the evaporation-induced apparent contact angles. The microregion is shaped by the effects of viscosity, Laplace and disjoining pressures (the latter in the form of an inverse-cubic law), and evaporation. The evaporation is in turn determined by heat conduction across the liquid film, kinetic resistance, and the Kelvin effect (i.e., saturation-condition dependence on the liquid-vapor pressure difference). While an asymptotic limit of large kinetic resistances was considered by Morris nearly two decades ago [J. Fluid Mech. 432, 1 (2001)], here we are concerned rather with matched asymptotic expansions in the limits of weak and strong Kelvin effects. Certain extensions are also touched upon within the asymptotic analysis. These are a more general form of the disjoining pressure and account for the Navier slip. Most notably, these also include the possibility of Wayner's extended microfilms (covering macroscopically dry parts of the substrate) actually getting truncated. A number of isolated cases encountered in the literature are thereby systematically recovered.

Figure 1

Configuration studied: the evaporation-induced contact angle ${\theta }_{\mathrm{mic}}$ generated within the microregion of a contact line considered to be at rest. Two possible microstructure regimes are shown in the enlargements.

Figure 2

Auxiliary function defined by Eq. (37).

Figure 3

Variation of the evaporation-induced contact angle with the kinetic resistance number. Extended-microfilm (Wayner) regime and the limit of weak Kelvin effect, $E\ll 1$.

Figure 4

Variation of the evaporation-induced contact angle with the kinetic resistance number for a few values of the dimensionless slip length. Extended-microfilm (Wayner) regime and the limit of weak Kelvin effect, $E\ll 1$.

Figure 5

Variation of the evaporation-induced contact angle with the kinetic resistance number for a few values of the dimensionless spreading coefficient. Truncated-microfilm regime and the limit of weak Kelvin effect, $E\ll 1$. At $\mathrm{\Sigma }=1$, the results coincide with those for the extended-microfilm (Wayner) regime.

Figure 6

Variation of the evaporation-induced contact angle with the dimensionless spreading coefficient for a few values of the kinetic resistance number. The limit of weak Kelvin effect, $E\ll 1$. The solid lines correspond to the truncated-microfilm regime, existing for $0<\mathrm{\Sigma }<1$. The dashed lines are for the extended-microfilm (Wayner) regime (no dependence on $\mathrm{\Sigma })$. The dot-dashed line represents the universal behavior at small $\mathrm{\Sigma }$ for the truncated-microfilm regime.

Figure 7

Variation of the microregion contact angle (evaporation induced in its entirety for $\tilde{\mathrm{\Sigma }}\ge 0)$ with the dimensionless spreading coefficient in the asymptotic domain $\tilde{\mathrm{\Sigma }}=O\left(1\right)$. Truncated-microfilm regime and the limit of weak Kelvin effect, $E\ll 1$.

Figure 8

Variation of the evaporation-induced contact angle with the kinetic resistance number at zero spreading coefficient $\left(\tilde{\mathrm{\Sigma }}=0\right)$. Truncated-microfilm regime and the limit of weak Kelvin effect, $E\ll 1$.

Figure 9

Variation of the evaporation-induced contact angle with the kinetic resistance number. Perfect wetting and the limit of strong Kelvin effect, $E\gg 1$.