Figure 1

The effective interface potential $\varphi \left(z\right)$ (full red line) according to Eq. (4) and the corresponding disjoining pressure $\Pi \left(z\right)=-{\varphi }^{\prime }\left(z\right)$ (dashed blue line) in units of ${\varphi }_0$ and ${\varphi }_0/h_0$, respectively. The positions of the minimum of $\varphi \left(z\right)$ at $z=h_0$ and of its inflection point at $z=h_i\equiv \sqrt[6]{3}{h}_0\approx 1.2h_0$ are indicated by vertical dotted lines. Also shown is ${\varphi }^{\prime \prime }\left(z\right)$ (dash-dotted green line), which appears in the second variation operator ${\widehat{O}}_h$ in Eq. (10) and which determines the stability of flat film solutions [see Eq. (17)]:Reuse & Permissions

Figure 2

Interface configurations $h^{\left(\text{eq}\right)}$ of nanodroplets as obtained by numerical minimization of the functional $\mathcal{F}$ in Eq. (3) based on the effective interface potential in Eq. (4) for $A/\left(\pi h_0^2\right)={100}^2$ and under the constraint $V_{\mathrm{ex}}/\left(A{h}_0\right)=0.5$, with ${\varphi }_0/\sigma =0.5$ [${\theta }_{\mathrm{eq}}={60}^{\circ }$, left panel (a)], and ${\varphi }_0/\sigma =0.1$ [${\theta }_{\mathrm{eq}}={26}^{\circ }$, right panel (b)]. In the projected side view (bottom row), the underlying substrate of area $A$ is indicated in blue. Lengths are given in units of $h_0$. Drop heights are measured from above the wetting film thickness $h_w^{\left(\text{eq}\right)}=1.009h_0$ in (a) and $h_w^{\left(\text{eq}\right)}=1.012h_0$ in (b). During the iterative minimization process, the mesh size of the triangulation has been coupled to the evolution of the interface shape in an adaptive way in order to optimize the spatial resolution locally. The lateral boundary conditions are implemented by a constraint on the boundary vertices, such that their lateral coordinates are fixed during the minimization process while the perpendicular height coordinate can evolve freely, effectively corresponding to neutral wetting (contact angle ${90}^{\circ }$) at vertical side walls (not shown) or Neumann boundary conditions.Reuse & Permissions

Figure 3

Vertical cut through the apex of a fully numerically obtained interface profile $h^{\left(\text{eq}\right)}$ (red squares) and the corresponding approximate profile $h_c^{\left(\text{eq}\right)}$ (full blue line) consisting of a spherical cap resting on a flat film. The profiles correspond to $A/\left(\pi h_0^2\right)={100}^2$, $V_{\mathrm{ex}}/\left(A{h}_0\right)=0.5$, and ${\varphi }_0/\sigma =0.5$, which are the parameters corresponding to Fig. 2a and to the bottom line in Table . Although the domain for the numerical calculation is rectangular, the droplet shape is to a good approximation radially symmetric ($r=\sqrt{x^2+y^2}$). The wetting film height for both profiles is $h_w^{\left(\text{eq}\right)}=1.01h_0$ and the contact angle is ${\alpha }_{\mathrm{eq}}={55}^{\circ }$ compared with ${\theta }_{\mathrm{eq}}={60}^{\circ }$ for the corresponding macroscopic drop. The free energies for these profiles agree up to the third digit.Reuse & Permissions

Figure 4

The approximate interfacial free energy $F(\alpha ,h_w)$ close to the macroscopic limit as described by Eq. (23): $A/\left(\pi h_0^2\right)={10}^8$, $V_{\mathrm{ex}}/\left(A{h}_0\right)=0.06$, and for ${\varphi }_0/\sigma =0.5$ corresponding to ${\theta }_{\mathrm{eq}}={60}^{\circ }$. $F_f=A[\sigma +\varphi \left(h_f\right)]$ is the free energy of the flat film solution for these parameters; $h_f-h_0=V_{\mathrm{ex}}/A$. The global minimum is located at $\alpha \approx {\theta }_{\mathrm{eq}}={60}^{\circ }$ and $h_w\approx h_0$. The contour lines are almost parallel to the $\alpha$ axis. (The contour lines shown range from $0.9792$ to $0.98$ in steps of $0.0001$ and from $0.98$ to $1.0$ in steps of $0.001$.) The inset shows the equilibrium angle ${\alpha }_{\mathrm{eq}}$ upon approaching the macroscopic limit as described by Eq. (23) for the same excess volume as used in the main figure; ${\alpha }_{\mathrm{eq}}$ approaches ${\theta }_{\mathrm{eq}}$ from below. The error bars are due to numerical inaccuracies. With $h_f/h_0=V_{\mathrm{ex}}/\left(A{h}_0\right)+1=1.06<h_i/h_0=1.2$ [see Eq. (18)] the flat film solution with $(h_w-h_0)/(h_f-h_0)=1$ is expected to be linearly stable (i.e., metastable). But the number of data points calculated here is too small in order to be able to detect the corresponding free energy barrier (cf. Fig. 9 for a smaller substrate).Reuse & Permissions

Figure 5

The pressure $p=-\Pi \left(h_w^{\left(\text{eq}\right)}\right)$ [see Eq. (22)] [in units of $p_f=-\Pi \left(h_f\right)$] calculated from the approximate free energy in Eq. (21) as a function of the substrate size $A$ and of the excess volume $V_{\mathrm{ex}}$ for (a) $V_{\mathrm{ex}}/\left(A{h}_0\right)=0.04$ and (b) $V_{\mathrm{ex}}/\left(A{h}_0\right)=0.25$, i.e., for a fixed homogeneous film thickness $h_f=1.04h_0<h_i$ and $h_f=1.25h_0>h_i$, respectively, and ${\varphi }_0/\sigma =0.5$. Note that for a fixed ratio $V_{\mathrm{ex}}/\left(A{h}_0\right)=v_{\text{ex}}$ one has $V_{\mathrm{ex}}/h_0^3=\pi v_{\text{ex}}{x}^2$ with $x={[A/\left(\pi h_0^2\right)]}^{1/2}$. Open blue and full green symbols indicate metastable and stable states, respectively, and red circles with crosses indicate unstable droplet solutions. Stable and metastable flat film solutions are indicated by boxes and stable and metastable droplets by circles. The vertical line indicates the morphological transition between stable films and stable droplets as obtained via numerical comparison of the corresponding two free energies. (A Maxwell construction for determining this transition point is discussed in, e.g., Fig. 11.) $A\to \infty$ corresponds to the macroscopic limit. (a) For $h_f<h_i$, as in the present case corresponding to $V_{\mathrm{ex}}/\left(A{h}_0\right)=0.04$, the flat film solution is stable or metastable for all $A$. Droplets (lowest branch) occur for $\sqrt{A/\left(\pi h_0^2\right)}\gtrsim 140$ and they are stable for $\sqrt{A/\left(\pi h_0^2\right)}\gtrsim 180$. (b) Within the present free energy approximation, the flat film solution is stable or metastable [although it should be unstable according to Eq. (18)], and there is an unphysical unstable branch of droplet solutions (top branch). Droplets are stable or metastable for for all substrate sizes $A$.Reuse & Permissions

Figure 6

The approximate interfacial free energy $F(\alpha ,h_w)$ for $A/\left(\pi h_0^2\right)={100}^2$, $V_{\mathrm{ex}}/\left(A{h}_0\right)=0.10$, and ${\varphi }_0/\sigma =0.5$. $F_f=A[\sigma +\varphi \left(h_f\right)]$ is the free energy of the flat film solution for these parameters; $h_f-h_0=V_{\mathrm{ex}}/A$. The global minimum representing a stable nanodroplet is located at $(h_w^{\left(\text{eq}\right)}-h_0)/(h_f-h_0)\approx 0.2$ and ${\alpha }_{\mathrm{eq}}\approx {57}^{\circ }$, i.e., close to but smaller than the macroscopic equilibrium contact angle ${\theta }_{\mathrm{eq}}={60}^{\circ }$ for this system. The flat film solution $h_w^{\left(\text{eq}\right)}=h_f$ [so that $(h_w^{\left(\text{eq}\right)}-h_0)/(h_f-h_0)=1$] is metastable; for this solution there is no dependence on $\alpha$. Contour lines are shown in the range $0.9802$ to $0.9809$ in steps of $0.0001$ and from $0.981$ to $1.005$ in steps of $0.001$.Reuse & Permissions

Figure 7

The approximate free energy $F(\alpha ,h_w)$ as defined in Eq. (21) for $A/\left(\pi h_0^2\right)={100}^2$, $V_{\mathrm{ex}}/\left(A{h}_0\right)=0.06$, and ${\varphi }_0/\sigma =0.5$ (i.e., for the same parameters as in Fig. 6 but for a smaller value of $V_{\mathrm{ex}}$). $F_f=A[\sigma +\varphi \left(h_f\right)]$ is the free energy of the flat film solution for these parameters; $h_f-h_0=V_{\mathrm{ex}}/A$. The contact angle corresponding to the global minimum is ${\alpha }_{\mathrm{eq}}\approx {55}^{\circ }$ at $(h_w^{\left(\text{eq}\right)}-h_0)/(h_f-h_0)\approx 0.04$, i.e., smaller than in Fig. 6. The flat film solution $h_w^{\left(\text{eq}\right)}=h_f$ [so that $(h_w^{\left(\text{eq}\right)}-h_0)/(h_f-h_0)=1$] is metastable and exhibits no dependence on $\alpha$. Contour lines are shown in the range $0.99902$ to $0.9991$ in steps of $0.00002$ and from $0.9992$ to $1.003$ in steps of $0.0002$.Reuse & Permissions

Figure 8

The approximate interfacial free energy $F(\alpha ,h_w)$ for $A/\left(\pi h_0^2\right)={100}^2$, $V_{\mathrm{ex}}/\left(A{h}_0\right)=0.048$, and ${\varphi }_0/\sigma =0.5$ (i.e., for the same parameters as in Figs. 6 and 7 but for an even smaller value of $V_{\mathrm{ex}}$). $F_f=A[\sigma +\varphi \left(h_f\right)]$ is the free energy of the flat film solution for these parameters; $h_f-h_0=V_{\mathrm{ex}}/A$. For this excess volume the droplet solution has disappeared and the flat film solution $h_w^{\left(\text{eq}\right)}=h_f$ [so that $(h_w^{\left(\text{eq}\right)}-h_0)/(h_f-h_0)=1$] is the global minimum. Contour lines are shown in the range $0.999$ to $1.006$ in steps of $0.00025$.Reuse & Permissions

Figure 9

The minimal free energy $F_{\text{min}}\left(h_w\right)={\min }_{\alpha }F(\alpha ,h_w)$ as a function of the wetting film thickness $h_w$ for $A/\left(\pi h_0^2\right)={100}^2$, ${\varphi }_0/\sigma =0.5$, and several values of $V_{\mathrm{ex}}/\left(A{h}_0\right)$ ranging from $0.048$ (see Fig. 8) to $0.3$. $F_f=A[\sigma +\varphi \left(h_f\right)]$ is the free energy of the corresponding flat film solution for these parameters. The morphological transition between flat films and nano-droplets occurs between $V_{\mathrm{ex}}/\left(A{h}_0\right)=0.058$ and $0.056$. For $V_{\mathrm{ex}}/\left(A{h}_0\right)>0.2$ the flat film solution appears to becomes unstable as expected from Eq. (18) (in the inset see the enlarged view of the region near $h_w=h_f$). The symbols indicate the points calculated numerically. For $V_{\mathrm{ex}}/\left(A{h}_0\right)=0.3$ a small barrier cannot be ruled out on the basis of the available numerical data.Reuse & Permissions

Figure 10

(a) The free energy (normalized to the free energy $F_f\left(V_{\mathrm{ex}}\right)$ of the flat film solution) and (b) the pressure $p$ in units of $\sigma /h_0$ as a function of $V_{\mathrm{ex}}/\left(A{h}_0\right)$ for a fixed substrate size, $A/\left(\pi h_0^2\right)={100}^2$, and ${\varphi }_0/\sigma =0.5$ as obtained from the approximate free energy expression $F(\alpha ,h_w)$. Global minima (full green), local minima (open blue), and saddle points or local maxima (symbols with red crosses) are shown. The upper branch in (b) corresponds to flat films (boxes) and the lower one to nanodroplets (circles). Within the reduced model we have $p=-\Pi \left(h_w^{\mathrm{eq}}\right)$ [see Eq. (22)]. The pressure values obtained from a numerical minimization of Eq. (3) [black diamonds; $p\approx -\Pi \left(h_w^{\mathrm{eq}}\right)$; see Eq. (20)] agree well with the results obtained from the approximate free energy. The dashed vertical line indicates the volume at which $h_f=h_i$. At this volume the unstable droplet branch merges with the flat film branch in (a) as well as in (b). The full vertical line indicates the morphological transition between the film and the droplet configurations at which the free energies of the flat film solution (boxes) and of the (meta)stable droplet solution (circles) are equal [see (a)].Reuse & Permissions

Figure 11

Sketch of the Maxwell construction leading to the position of the thin full vertical line in Fig. 10. The color code corresponds to the one in Fig. 10: green, blue, and red indicate stable, metastable, and unstable states, respectively. $V_0$ denotes the volume of a film of thickness $h_0$, $V_m$ is the minimal volume required to form a droplet, $V_{\text{eq}}$ is the volume at which the free energies of the flat film and of the stable droplet are equal. For $V\nearrow V_i$ the branch of metastable flat films turns into a branch of unstable flat films. There also the branch of unstable droplets merges into the flat film branch.Reuse & Permissions

Figure 12

The droplet volume $V_d=V-A{h}_w$ as function of $V_{\mathrm{ex}}/\left(A{h}_0\right)$ (i.e., as a function of $V=V_{\mathrm{ex}}+A{h}_0$) for ${\varphi }_0/\sigma =0.5$ and (a) $A/\left(h_0^2\pi \right)={100}^2$ and (b) $A/\left(h_0^2\pi \right)={1000}^2$ as obtained from the approximate expression $F(\alpha ,h_w)$ for the free energy: for fixed $V=V_{\mathrm{ex}}+A{h}_0$ the free energy landscapes (see, e.g., Figs. 6, 7, 8) have been calculated and the wetting film thicknesses $h_w$ of the droplet solutions, if they exist, have been determined. The upper branch corresponds to the stable (green) or metastable (blue) droplet solution. For (a) this is the lower branch in Fig. 10. The points on the abscissa correspond to stable (green) or metastable (blue) flat film solutions. The comparison between (a) and (b) shows that the minimal droplet size $V_d^c$ (for which the unstable and the metastable droplet branches meet) increases upon increasing the substrate area while the corresponding excess volume $V_{\mathrm{ex}}^c$ in units of the substrate area decreases.Reuse & Permissions

Figure 13

(a) Disjoining pressure $\Pi \left(h_w\right)$ with $h_0<h_w<h_f$ [calculated for the model potential in Eq. (4)] in the wetting film (solid red line) and the right-hand side of Eq. (27) with $R(h_w,h_f,A)$ from Eq. (29) (dashed blue lines) for $A/\left(\pi h_0^2\right)={10}^4$ and ${\varphi }_0/\sigma =0.5$ as a function of the wetting film thickness $h_w$ for $h_f=1.04h_0$, $h_f=h_f^c=1.05005h_0$, and $h_f=1.06h_0$, which fixes $V=A{h}_f$ for a given $A$ [see Eq. (12)]. We also show the right-hand side of Eq. (27) for $A/\left(\pi h_0^2\right)={10}^5$ and $h_f=h_f^c=1.02587h_0$ (dash-double-dotted cyan line), as well as for $A/\left(\pi h_0^2\right)={10}^6$ and $h_f=h_f^c=1.01394h_0$ (dash-dotted green line). The thin dotted red line shows the linear fit to $\Pi \left(h_w\right)$ at $h_w=h_0$. For $h_f=h_f^c\left(A\right)$ the curves $\Pi \left(h_w\right)$ and the one for the right-hand side of Eq. (27) touch each other at a single point at $h_w=h_w^c\left(A\right)$ indicated by a magenta circle. (b) $h_f^c$ and $h_w^c$ as a function of the substrate area $A$ in units of $\pi h_0^2$ as obtained graphically from (a) (black squares and magenta circles, respectively) and from the analytic approximation described in the main text (full black and dotted magenta line, respectively).Reuse & Permissions