Figure 1

Evolution snapshots of $D\times L$ thin films with $D=50,L=512$, for four times ($t=180,1800,18000$ and $1800000$, respectively). We show results for ${\Psi }_g=0.2$ (a) and ${\Psi }_g=0.6$ (b). Regions with $\Psi (x,z,t)>0$ are shown in black, and regions with $\Psi (x,z,t)<0$ are shown in white. We notice that the last snapshot in (a) exhibits two AB interfaces of length $D$, whereas the final snapshot in (b) exhibits a single interface of length $L$.Reuse & Permissions

Figure 2

(a) Graphical illustration of Young's equation, showing $\cos \theta$ as a function of $({\gamma }_{wv}-{\gamma }_{wl})/{\gamma }_{vl}$, indicating the regimes of complete and incomplete drying, as well as incomplete and complete wetting, respectively. (b) The equilibrium state [for the case where both coexisting phases, vapor ($v$) and liquid ($l$), occupy a volume fraction of $50\%$ of the strip] contains an interface between the coexisting phases. On average, this runs perpendicular to the walls ($w$) at $x\simeq L/2$, irrespective of whether wetting/drying is complete or incomplete. For $({\gamma }_{wv}-{\gamma }_{wl})/{\gamma }_{vl}<-1$ (complete drying), wall-liquid contact is avoided because of a thin drying layer, which leads to a cost of $2{\gamma }_{vl}(L/2)$ of vapor-liquid interfacial free energy, in addition to the free energy cost of the vapor at the two walls, $2{\gamma }_{wl}L$. Thus, the total free energy cost is $(2{\gamma }_{wv}+{\gamma }_{vl})L$. (c) However, if one considers a layered structure [left panel in (c)], the two vapor-liquid interfaces extend over $L$ instead of $L/2$ only. Hence, the total free energy cost is $2({\gamma }_{wv}+{\gamma }_{vl})L$. Likewise, the layered structure occurring under conditions of complete wetting [right frame of (c)] has an interfacial free energy cost $2({\gamma }_{wl}+{\gamma }_{vl})L$. This exceeds the free energy cost of the equilibrium structure in (b), which is $(2{\gamma }_{wl}+{\gamma }_{vl})L$. In comparison with ${\gamma }_{vl}L$, the cost of the perpendicular interface ($\sim {\gamma }_{vl}D$) is always neglected, since $L\gg D$.Reuse & Permissions

Figure 3

Order parameter profiles, ${\Psi }_{\mathrm{av}}(z,t)$ vs. $z$, averaged over the $x$ direction parallel to the boundaries. We show results for $D=50,L=512$, and ${\Psi }_g=0.2$ (a), ${\Psi }_g=0.6$ (b). Five times $t$ are shown, as indicated.Reuse & Permissions

Figure 4

Normalized correlation functions $C(x,z,t)/C(0,z,t)$ vs. $x/\ell (z,t)$ for ${\Psi }_g=0.2$ and $z=10$ (a), and $z=D/2=25$ (b). Panel (c) shows the time dependence of the length $\ell (z,t)$ for various layers, as indicated. Panel (d) shows $\ell (z=0,t)$ for several values of ${\Psi }_g$.Reuse & Permissions

Figure 5

Early-time results for the case with no noise ($\epsilon =0$). (a) Laterally averaged profiles for $t=6,9$, reminiscent of SDSD waves. The parameters are ${\Psi }_g=0.1$ and $D=50$. (b) Plot of the first intersection of ${\Psi }_{\mathrm{av}}(z,t)$ with ${\Psi }_{\mathrm{in}}\left(z\right)$, $R_1\left(t\right)$ vs. $t$, for the profiles corresponding to $D=50$ and ${\Psi }_g=0.02,0.05,0.1$.Reuse & Permissions