Viscoelastic wetting transition: beyond lubrication theory

Abstract

The dip-coating geometry, where a solid plate is withdrawn from or plunged into a liquid pool, offers a prototypical example of wetting flows involving contact-line motion. Such flows are commonly studied using the lubrication approximation approach which is intrinsically limited to small interface slopes and thus small contact angles. Flows for arbitrary contact angles, however, can be studied using a generalized lubrication theory that builds upon viscous corner flow solutions. Here we derive this generalized lubrication theory for viscoelastic liquids that exhibit normal stress effects and are modelled using the second-order fluid model. We apply our theory to advancing and receding contact lines in the dip-coating geometry, highlighting the influence of viscoelastic normal stresses for contact line motion at arbitrary contact angle.

Fig. 1

Dip-coating geometry. The interface can be described by h(s) and $\theta (s)$. The plate is being pulled at an angle ${\theta }_p$ (here $\pi /2$) with respect to the stationary fluid bath. The interface depth from the contact line to the bath is denoted by $\mathrm{\Delta }$, which on the scale of the meniscus has an apparent (outer) angle ${\theta }_{\text{app},o}$. The microscopic angle at the contact line is ${\theta }_e$, as shown in the zoomed picture

Fig. 2

Corner flow geometry: (a) Definition of the polar coordinates r and $\varphi$ in the locally tangent wedge of angle $\theta (s)$. At this tangent position s along the interface, we use the corner flow solution (see panel (b)). Here, the interface point corresponds to $\varphi =0$ and the plate is at $\varphi =\theta (s)$. (b) Illustration of the streamlines in the corner flow solution, in a wedge of constant angle $\theta =\theta (s)$, sketched in the frame comoving with the interface

Fig. 3

Phenomenology of viscoelastic dip-coating obtained by numerically integrating equation (39), illustrated for ${\theta }_p={90}^{\circ },{\theta }_e={90}^{\circ }$, and ${\lambda }_s/{\ell }_{\gamma }={10}^{-4}$. (a) Typical interface profiles for advancing motion ($\text{Ca}<0$, green and orange curves) and receding motion ($\text{Ca}>0$, brown and purple curves). The right side displays the Newtonian solutions for $N_0=0$, while the viscoelastic solutions (left) are plotted for $N_0=1\times {10}^3$. (b) Vertical position of the contact line $\overline{\mathrm{\Delta }}=\mathrm{\Delta }/{\ell }_{\gamma }$ versus capillary number $\text{Ca}$, for different values of the viscoelastic material parameter $N_0$. The crosses correspond to the interface profiles of figure (a) using the same color code. The vertical dashed line represents the critical capillary number for $N_0=0$, beyond which no solution exists as a Landau–Levich film is entrained. The solutions in the upper branch beyond $\mathrm{\Delta }/{\ell }_{\gamma }=\sqrt{2}$ will not be discussed in this work. In the viscoelastic case, the numerical integration of (39) cannot be achieved for Ca below a certain value, as discussed in Sect. 4.3

Fig. 4

Receding contact lines, for small angles ${\theta }_p={10}^{\circ },{\theta }_e={10}^{\circ }$, and ${\lambda }_s/{\ell }_{\gamma }={10}^{-4}$. (a) Contact line position $\overline{\mathrm{\Delta }}/{\theta }_p=\mathrm{\Delta }/\left({\theta }_p,{\ell }_{\gamma }\right)$ versus capillary number $\text{Ca}$. (b) Critical capillary number ${\text{Ca}}_c$ as a function of the viscoelastic material parameter $N_0$. The blue circles are obtained by numerically integrating generalized lubrication equation (39). The green crosses are the numerical solutions of the lubrication equation (37). The modified Cox-Voinov represents the asymptotic expansion (47) using (48)

Fig. 5

Receding contact lines, for ${\theta }_p={90}^{\circ },{\theta }_e={90}^{\circ }$, and ${\lambda }_s/{\ell }_{\gamma }={10}^{-4}$. (a) Contact line position $\overline{\mathrm{\Delta }}=\mathrm{\Delta }/{\ell }_{\gamma }$ versus capillary number $\text{Ca}$. (b) Critical capillary number ${\text{Ca}}_c$ as a function of the viscoelastic material parameter $N_0$. Similarly to Fig. 4, blue circles are numerical solutions of (39), while the orange dashed line represents the modified Cox-Voinov prediction corresponding to (47) using (49)

Fig. 6

Receding contact lines with varying wettability of the plate, ${\theta }_e={30}^{\circ },{60}^{\circ },{90}^{\circ }\&{150}^{\circ }$, for fixed values ${\lambda }_s/{\ell }_{\gamma }={10}^{-4},{\theta }_p={90}^{\circ }$. (a) Critical capillary number ${\text{Ca}}_c$ versus Weissenberg number $\text{Wi}$. (b) Rescaled critical capillary number ${\text{Ca}}_c/{\theta }_e^3$ versus Weissenberg number $\text{Wi}$. (c) Rescaled critical capillary number ${\text{Ca}}_c/{\theta }_e^3$ versus viscoelastic material parameter $N_0$

Fig. 7

Advancing contact lines, for small angles ${\theta }_p={10}^{\circ },{\theta }_e={10}^{\circ }$, and ${\lambda }_s/{\ell }_{\gamma }={10}^{-4}$. (a) Contact line position $\overline{\mathrm{\Delta }}/{\theta }_p=\mathrm{\Delta }/\left({\theta }_p,{\ell }_{\gamma }\right)$ versus capillary number $\text{Ca}$. The local minima and the inflection points are represented by orange circles and red crosses respectively. The dashed lines represent the predictions for the weakly viscoelastic regime (50), using the relation between $\mathrm{\Delta }/({\theta }_p{\ell }_{\gamma })$ and ${\theta }_{\text{app},o}$ from (51). (b) Absolute values of the capillary number at the local minimum of $\overline{\mathrm{\Delta }}/{\theta }_p$ and at the inflection points, plotted as a function of the viscoelastic material parameter $N_0$. The dashed lines are the asymptotic predictions, given by (52) and (55)

Fig. 8

Advancing contact lines, for small angles ${\theta }_p={10}^{\circ },{\theta }_e={10}^{\circ }$, and ${\lambda }_s/{\ell }_{\gamma }={10}^{-4}$, for a selected value of $N_0=1\times {10}^3$. The two dashed lines represent the predictions for the weakly viscoelastic regime (50) and the strongly viscoelastic regime (53), using the relation between $\mathrm{\Delta }/({\theta }_p{\ell }_{\gamma })$ and ${\theta }_{\text{app},o}$ from (51)

Fig. 9

Advancing contact lines, for ${\theta }_p={90}^{\circ },{\theta }_e={90}^{\circ }$, and ${\lambda }_s/{\ell }_{\gamma }={10}^{-4}$. (a) Contact line position depth $\overline{\mathrm{\Delta }}=\mathrm{\Delta }/{\ell }_{\gamma }$ versus capillary number $\text{Ca}$. (b) Capillary number at the location of local minimum of $\overline{\mathrm{\Delta }}$ as a function of the viscoelastic material parameter $N_0$. The dashed lines in (a) and (b) represent the predictions for the weakly viscoelastic regime in (50) and (52) respectively