Wetting at nanoscale: Effect of surface forces and droplet size

Despite being intensively investigated, wetting at the nanoscale leaves a manifold of questions unresolved. In particular, the dependence of a contact angle on a droplet size for the droplets of the height of the order of a few nanometers is intensively debated. This effect is believed to be related to intermolecular (surface) forces. In the present work, we use the disjoining pressure concept and solve the Derjaguin equation numerically and analytically to model profiles of the sessile droplets of heights comparable with the range of the surface force action. We show that values of the contact angle are dramatically dependent on the droplet height as well as the way the contact angle is defined. For the axisymmetric droplets, the contact angle increases with increasing droplet height, and this dependency becomes universal for different disjoining pressure isotherms when plotted dimensionless with respect to the surface force action range. We demonstrate that for cylindrical droplets, different contact angle definitions can lead to opposite dependencies on the droplet size. We show as well that varying orders of magnitude of the apparent line tension reported can be additionally explained by the contact angle definition chosen.

Figure 1

The sessile droplet in equilibrium with its surroundings for the case of pseudopartial wetting. On the left side is the schematic representation of cylindrical and axisymmetric droplets. On the right side, the disjoining pressure as a function of the local droplet height alongside with the droplet profile is shown. The green points are as follows: $h_1$ is the first root of Eq. (8) corresponding to thickness of the equilibrium adsorbed/wetting layer for the liquid wedge, $h_{\mathrm{ads}}$ is the solution of the equation $\mathrm{\Pi }\left(h\right)+p_e\left(r\right)=0$ corresponding to the thickness of the equilibrium adsorbed layer for the small droplet, $h_2$ is the second root of Eq. (8) defining the range of the surface force action, and $h_m$ corresponds to the maximal slope (inflection point). In points A and M, the equation $\mathrm{\Pi }\left(h_{\mathrm{ads}}\right)=\mathrm{\Pi }\left(h_m\right)$ is fulfilled. The contact angle of the droplet is denoted as $\theta$.

Figure 2

Disjoining pressure isotherms corresponding to different cases in Table 1. Group A is shown in black, group B is shown in blue. (a) The region of the disjoining-conjoining transition. The red markers depict the location of ${\overline{h}}_1$. The red arrow is to exemplify the location of ${\overline{h}}_1$ for isotherm 3B. (b) The region of the second conjoining-disjoining pressure transition and the disjoining part of the isotherm responsible for the repulsion between interfaces. The red marker and arrow depict the location of ${\overline{h}}_2=1$.

Figure 3

(a) Stationary droplet profiles obtained from the Derjaguin equation (15) for different heights of droplets ${\overline{h}}_d$. Isotherm 2B has been used for the profiles depicted. (b) The dependency of the contact angle $\theta$ defined at the inflection point on the droplet height ${\overline{h}}_d$ is shown with teal squares. The contact angles evaluated using the spherical cap of the first kind (see discussion in the text) are marked with teal stars, the contact angle in the Xu-Salmeron model [Eq. (19); see discussion in the text] is shown as a solid blue line. The straight black solid line is to illustrate the contact angle ${\theta }_w$ of the liquid wedge or large droplet [Eq. (7)]. The blue triangles show the contact angles evaluated using the full Frumkin-Derjaguin equation (9). The blue arrow is to show that all teal/blue-marked dependencies belong to the left (angle) axis. The adsorbed/wetting film thickness ${\overline{h}}_{\mathrm{ads}}$ is shown with red squares. The equilibrium wetting film thickness ${\overline{h}}_1$ corresponding to the liquid wedge or large droplet is shown with the dashed red line. The red arrow is to show that all red-marked dependencies belong to the right (thickness) axis.

Figure 4

(a) The dependency of the contact angle ${\theta }_1$ on the droplet height ${\overline{h}}_d$ is marked by squares. The contact angles ${\theta }_2$ and ${\theta }_3$ estimated using spherical caps of the first and second kinds are marked by stars and circles, respectively. The lines are a guide to the eye. The dashed black line is to illustrate the contact angle ${\theta }_w$ of the liquid wedge or large droplet [Eq. (7)]. Three droplet profiles with different ${\overline{h}}_d$ have been chosen to show the differences in the profile geometry. Those correspond to panels (b), (c), and (d). The numerically obtained profiles are shown with black solid lines, the spherical caps of the first and second kinds with solid blue and dashed green lines, respectively. (b) The graphical means of the contact angle choice: ${\theta }_1$ denotes the contact angle defined at the inflection point, ${\theta }_2$ denotes the angle defined at the intersection of the spherical cap of the first kind with abscissa axis, and ${\theta }_3$ denotes the angle defined at the intersection of the spherical cap of the second kind with abscissa axis. All profiles correspond to Case 2B.

Figure 5

(a) Traction $\overline{T}\left(\overline{r}\right)$ (the curvature-induced pressure) distribution over the wetted area. The shape of $\overline{T}\left(\overline{r}\right)$ under the droplet shows the transition from “sinking” to the dimple emergence occurring as the droplet size increases. The region where the dimples appear is shown by arrows. (b) Pressure dimples emerging for the droplets of ${\overline{h}}_d>{\overline{h}}_d^{cr}$. (c) The disjoining pressure distribution along the height of the droplets for which no dimples emerge (solid black line) and dimples emerge (dashed blue line), accordingly. The inset plot is to schematically illustrate the droplet profiles and the disjoining pressure distribution. The plots correspond to the isotherm 2B.

Figure 6

The slopes $v$ for 2D (cylindrical) droplet obtained from Eq. (28) and 3D (axisymmetric) droplet obtained as a numerical solution of Eq. (15) shown with solid and dashed lines, accordingly, are plotted for ${\overline{h}}_d=1.59$ (${\overline{h}}_d$ corresponds to 3D droplet) on the left side (a), and for ${\overline{h}}_d=6.36$ (${\overline{h}}_d$ corresponds to 3D droplet) on the right side (b). The isotherm corresponding to the case 2B is used. The red dash-dotted line is to show the range of ${\overline{h}}_d$, within which the 2D model demonstrates good agreement with the 3D model.

Figure 7

(a) Contact angles for cylindrical droplets evaluated using different approaches. The dependency of the contact angle ${\theta }_1$ defined at the inflection point on the droplet height ${\overline{h}}_d$ is marked with squares. The contact angles evaluated using cylindrical caps of the first and second kinds are marked with stars and circles, respectively. The dashed black line is to illustrate the contact angle ${\theta }_w$ of the liquid wedge or large droplet [Eq. (7)]. The blue triangles show the contact angles evaluated using the full Frumkin-Derjaguin equation (9). The lines are a guide to the eye. Three examples of profiles and cylindrical caps (b), (c), and (d) show the differences in the profiles' geometry. The semianalytically obtained profiles are with black solid lines, and the cylindrical caps of the first and second kinds with solid blue and dashed green lines, respectively. The inset picture in (d) shows the magnified contact line region and departure of the cylindrical caps from the numerically evaluated profile. All profiles were plotted for isotherm 2B.

Figure 8

(a) Influence of the surface forces on the dependence of the contact angle on the dimensionless droplet height for the axisymmetric droplet. The contact angle is defined at the inflection point. (b) Dependence the dimensionless adsorbed/wetting film thicknesses on the dimensionless droplet height.

Figure 9

(a) Dependencies of the contact angle on the dimensionless height of the cylindrical (2D) droplet obtained with Eq. (44) (shown with empty black markers connected with dashed lines) as well as the contact angles resulting from the solution of the Derjaguin equation (28) (shown with filled colored markers). (b) The simplified disjoining pressure isotherm (solid blue line) where ${\overline{h}}_1$ is the dimensionless thickness of the equilibrium adsorbed/wetting film for the liquid wedge, ${\overline{h}}_{\mathrm{ads}}$ is the dimensionless thickness of the adsorbed/wetting film for the droplet, ${\overline{h}}_{\min }$ is the height at which the disjoining pressure attains its minimal value ${\overline{\mathrm{\Pi }}}_{\min }$, ${\overline{h}}_m$ corresponds to the inflection point, and ${\overline{h}}_2=1$ defines the range of the surface force action.

Figure 10

The dependency of the contact angle $\theta$ on the dimensionless curvature of the three-phase contact line $1/{\overline{r}}_d$ (a) for the contact angle defined at the inflection point and (b) for the contact angle defined at the intersection of the spherical cap of the first kind (three-point-based) with abscissa axis. Inset schematic pictures illustrate the way that the contact angle is defined. The solid red lines are a guide for the eye and schematically show the slopes required for the apparent line tension evaluation.