New analytical solutions for static two-dimensional droplets under the effects of long- and short-range molecular forces

Abstract

We report new analytical solutions for the thickness profile of partially wetting two-dimensional droplets. The model includes the effects of capillarity and both short- and long-range molecular forces. We analyze the dependence of the maximum thickness, the contact angle, and the cross-sectional area on the height of the nanometric precursor film that surrounds the droplet. We found asymptotic expressions for the thickness profile and for the contact angles for large and small droplets. The results are compared to those obtained previously for polar liquids. The analytical solutions found here are useful to assess the validity of the hypothesis and the semi-analytical solutions proposed in the literature. In addition, these solutions enable the inference of information about the molecular potential from the measured steady profiles.

Appendix: Relationship between the contact angles \(\varTheta _s\) and \(\varTheta \)

In this study, we establish the connection between the thermodynamic definitions of the contact angle $\varTheta _s$, Eq. (42) and the contact angle $\varTheta$ defined as the maximum slope of the dimensional profile (see Sect. 5.1). As we discussed in Sect. 6, $\varTheta \rightarrow \varTheta _s$ only in the limit of large droplets.

In the following, we return to the dimensional variables. Thus, Eq. (7) takes the form:

$$\begin{aligned} \gamma K = \frac{\text {d}U(h)}{\text {d}h} - P, \end{aligned}$$

(44)

where $K=h_{xx}/(1+h_{x}^2)^{3/2}$ is the curvature, and P is an unknown constant [10, 11]. Imposing that $K\rightarrow 0$ when $h \rightarrow h_\mathrm{f}$ (film region), the value of P is

$$\begin{aligned} P = -\varPi (h_\mathrm{f}). \end{aligned}$$

(45)

We now integrate Eq. (44) from $h_\mathrm{f}$ to an arbitrary thickness H:

$$\begin{aligned} \gamma \int _{h_\mathrm{f}}^H K \;\text {d}h + \int _{h_\mathrm{f}}^H \varPi (h) \, \text {d}h = -\int _{h_\mathrm{f}}^H P \, \text {d}h. \end{aligned}$$

(46)

The three integrals in (46) can be calculated to get

$$\begin{aligned} \gamma (1-\cos {\alpha (H)})= U(H)-U(h_\mathrm{f}) + (h_\mathrm{f}-H)P, \end{aligned}$$

(47)

where the angle $\alpha$ is defined as the angle between the substrate and the profile at any h, i. e., $\tan {\alpha }(h)=h_x$.

Equation (47) shows that the angle $\alpha$ at any thickness H will depend on H and U. If we used $K=h^{\prime \prime }(x)$, valid under the lubrication hypothesis, the left-hand side of Eq. (47) would read $\gamma /2\tan ^2\alpha (H)$.

Notice that the angle $\alpha$ evaluated at the point $h_\mathrm{c}$, where the slope is maximum, is the contact angle of the dimensional profile $\varTheta$. Then, to show that the thermodynamic contact angle $\varTheta _s$ is the limit of $\varTheta =\alpha (h_\mathrm{c})$ for large droplets, we first evaluate Eq. (47) in $H = h_\mathrm{c}$

$$\begin{aligned} \gamma (1-\cos {\varTheta })= U(h_\mathrm{c})-U(h_\mathrm{f}) + (h_\mathrm{f}-h_\mathrm{c})P. \end{aligned}$$

(48)

We now consider large droplets by taking the limit $h_\mathrm{f}\rightarrow h_*$. From Eqs. (45,31), we have $\{U(h_\mathrm{c});(h_\mathrm{f}-h_\mathrm{c})P\}\rightarrow \{0;0\}$. Denoting $\varTheta _*\equiv \varTheta (h_\mathrm{c}\rightarrow \infty )$ we conclude that, in this limit, Eq. (48) becomes

$$\begin{aligned} \gamma (1-\cos {\varTheta _*})= -U(h_*), \end{aligned}$$

(49)

which is the usual definition for the thermodynamic contact angle $\varTheta _s$ [47]. The conclusion is that $\varTheta _* \equiv \varTheta (h_\mathrm{c} \rightarrow \infty ) = \varTheta _s$, and then $\varTheta _s$ can only be observed in large droplets. As mentioned above, when the lubrication approximation is employed, the term $(1-\cos {\varTheta })$ is replaced by $1/2\tan ^2\varTheta$ as shown in Eq. (42). Moreover, from the analysis in Sect. 5.1, where we show that the maximum slope monotonically increases as $h_\mathrm{f}$ decreases, it is straightforward that the thermodynamic angle given in Eq. (49) is the upper limit of the observable contact angle $\varTheta$, as discussed in the Conclusions.

From Eq. (49), we may also relate the strength of the disjoining pressure $\kappa$ with $\varTheta _s$, n, m, and $\gamma$. Effectively, since

$$\begin{aligned} U(h_*)=-\kappa \frac{h_*(n-m)}{(n-1)(m-1)}, \end{aligned}$$

(50)

then

$$\begin{aligned} \kappa =\frac{\gamma (1-\cos {\varTheta _*})(m-1)(n-1)}{h_* (n-m)}. \end{aligned}$$

(51)

Again, if the curvature is approached as $h_{xx}$, the factor $1-\cos {\varTheta _*}$ is replaced by $1/2\tan ^2{\varTheta _*}$. In the References the interested reader will find many examples where this relationship between the contact angle and $\kappa$ is employed. Remarkably, there is a simple relationship between the values of $\kappa$ and $h_*$ with the Hamaker constants of the molecular forces. If the disjoining–conjoining pressure is written as $\varPi =A_n/h^n-A_m/h^m$, where $A_i$ is related to the Hamaker constant, then

$$\begin{aligned} h_*=\left( \frac{A_n}{A_m}\right) ^{1/(n-m)} \end{aligned}$$

(52)

and

$$\begin{aligned} \kappa =\left( \frac{A_m^{\;\;n}}{A_n^{\;\;m}}\right) ^{1/(n-m)}. \end{aligned}$$

(53)

which is explained in the Appendix of Ref. [10].